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As an example, take noun classes similar to Swahili or Navajo. In Swahili each noun belongs to one class (accepting the same noun might change class for pluralisation). In Navajo, some classes are applied based on shape (long, flat etc).
Can you say a language still has a noun class system if a noun can take multiple class indicators at the same time? Like something that is long and flat getting the class indicators for both the long-class and the flat-class? Or is it something else?
For a Bantu example, in (some types of) Kinshasa Lingála the noun class system has collapsed into a three-way distinction between "singular" (1), "plural" (2), and "inanimate" (7).
So when speaking other varieties that preserve the full class system, some Kinshasa speakers now seem to consider "plural" a completely separate category from animacy/class, leading to multiple class markings on a single noun:
ma-i
6-water
"water" (a mass noun)
ba-ma-i
2-6-water
"waters" (that is, multiple separate sources of water, treated as a count noun)
Even in these special circumstances, though, this sort of double-marking is not common. "Noun class" generally acts like a single property of the noun, which can have only a single value, and this value is copied to other words to mark agreement. (In syntax, this property is often called the "φ feature", and the agreement "φ agreement".) When there are multiple values involved, they tend to indicate different orthogonal things (like person vs number vs gender), rather than multiple values that all indicate the object's shape.
Note after posting this answer: It appears I did not understand what the original poster was asking for. As a comparative (cross-linguistic) category, "noun class" is most often defined in terms of agreement: specifically, the presence of agreement markers on a "target" of grammatical agreement (often a grammatically subordinate word, such as a modifier), not by the morphology of the "controller" of agreement (e.g. the head noun in a noun phrase).
Taking Spanish as an example, although head nouns frequently have morphology or phonology that correlates to their noun class (masculine-class nouns often end in -o, feminine-class nouns often end in -a), this is not the defining feature of what makes a noun belong to one class or another, so we have nouns like problema, where despite ending in a, we can demonstrate that it is masculine by showing that in phrases like "unm problema seriom" the targets of gender agreement consistently take masculine forms.
However, what the question in fact seems to be asking for is languages where a morpheme that usually marks one noun class and a morpheme that usually marks another noun class can cooccur on a single word.
That question has no inherent connection with what I was thinking of: whether a noun triggers multiple types of noun class agreement on targets of agreement; i.e. with whether a noun shows syntactic behavior that would motivate categorizing it as belonging to "multiple noun classes" at the same time.
Regardless, I'll leave the answer in case other people interpreted the question the same way that I did.
In Spanish, the singular definite article usually has the form el for masculine nouns and la for feminine nouns, but some feminine nouns such as agua take the definite article el.
Based on the situations where this occurs, it seems like originally, this el was a phonologically-conditioned alternative form of the feminine definite article (all of the nouns in question start with a stressed /a/ sound, so the use of el instead of la avoids an /aa/ sequence), and prescriptively, this phenomenon doesn't affect words other than la, una, alguna, ninguna when positioned immediately before this kind of noun. But, since the rule as it exists in present-day Spanish is not fully predictable based on the pronunciation of a word and has many complicated conditions and exceptions (e.g. it doesn't apply to nouns starting in unstressed /a/, or to adjectives or proper nouns even when they start in stressed /a/), it seems it's not uncommon for Spanish speakers to reinterpret the rule for nouns in this class as "any preceding modifier takes masculine agreement".
However, modifiers following the noun still take feminine agreement.
So you can get cases like these given in jacobo's answer on Spanish Stack Exchange to the question "Why is "agua" masculine in singular form and feminine in plural? "El agua" / "Las aguas" ¿Por qué decimos "el agua" si es una palabra femenina?":
...elm abundante agua caídaf...
nuestrom habla riojanaf...
(These examples are taken from El abundante agua fría: Hermaphroditic Spanish Nouns by David Eddington and Jose Ignacio Hualde, (2008))
It seems like Swahili, mentioned in the question, can also provide examples of nouns with mixed agreement patterns.
The Wikipedia article "Swahili grammar" says
Animate nouns (i.e. those referring to people or animals) which are not in classes 1/2 generally take the agreement prefixes (concords) from classes 1/2 as if they did belong to it.
[...]
Animate nouns in classes 9/10 may exhibit a slight aberration from this pattern. The genitive pronominal forms -angu, -ako, -ake, -etu, -enu and -ao are frequently inflected with a group of nouns referring to close human relationships with their appropriate class 9/10 concords, regardless of the fact that they are animate (giving yangu, yako, yetu etc. in singular and zangu, zako, zetu etc. in plural). For some speakers, the same rule applies to the simple genitive preposition -a (giving ya in singular and za in plural), however for most speakers wa is used for all animate nouns regardless of number or class. Other parts of speech are unaffected by this exception.
Can you say a language still has a noun class system if a noun can take multiple class indicators at the same time? Like something that is long and flat getting the class indicators for both the long-class and the flat-class? Or is it something else?
My answer is yes.
In French, the plural word gens ("people") in general" is unique. It is clearly feminine diachronically and still demands feminine agreement for immediately preceding adjectives where there is a phonological difference and for associated quantifiers belonging to the same word group preceding the word gens. Anything not immediately preceding gens or not grouped with other preceding words are treated as masculine in keeping with the general rule in French that words generally applying to mixed genders are masculine. The word is also treated uniformly as masculine when followed by the preposition de (of) and a name of a quality or profession. Accordingly, you have sentences where gens can require different genders, sometimes within the same sentence. The rules are so unusual and so complex that even good native-speaking writers sometimes screw them up. Here are some examples of "good" usage that I think most well educated speakers would spontaneously use:
Tous(m) les gens sont là (All the people are there)
Toutes(f) les bonnes (f) gens sont là (All the good people are there)
Toutes (f) les bonnes (f) gens sont heureux (m) (All the good people are ; happy
Intruits (m) par l'expérience, les vielles (f) gens sont soupçonneux (m). Instructed by experience, the old people are very suspicious.
Certains (m) gens d'affaires
Certain business people
Nevertheless, gender in French is generally quite regular. If I replaced "gens" ("people") with "personnes" ("persons") in all the above examples, all the related words would be uniformly feminine, with perhaps a big change in authenticity, but little or no real difference in meaning.
Because of these strange rules, you have to say that gens, uniquely among French nouns, is both grammatically feminine and masculine at the same time when used in the plural. (On the rare occasions it is used in the singular, I think it is uniformly feminine.) The rest of the language does have its occasional quirks, but generally has a clean distinction between genders.
In German, which has masculine, feminine, and neuter nouns, there are a few common nouns, such as das Mädchen (the girl) whose neuter gender goes against "natural" gender. In such cases, agreement is strict within the noun phrase and generally within the sentence, but long distance agreement between sentences usually follows the natural gender in informal speech, so that the pronoun used in a following sentence to refer to the word Mädchen would be a form of sie (she)(feminine), rather than a form of es (it)(neuter) in violation of the formal rules of gender agreement.
In standard Swahili, in areas whereas it has always been natively spoken, I think that noun classes generally follow strict rules of concord that are metaphorically motivated in complex historical ways; however, in urban areas where it is now natively spoken and as an L2 language, nouns generally require mixed concord according to rules of animacy that apply to some parts of speech and some word classes, but not to others. Here are adapted examples taken from Wikipedia and my perhaps faulty memory (I have put the class number after the relevant Swahili word and tried to hyphenated the obvious Swahili agreement morphemes and to provide a literal English translation to make the morphemes clearer):
wa-naume (cl2) wa-kubwa (cl2) wa-li-anguk(a) (cl2) (normal concord)
(plural human) man big, they in the past fall (The big men fell)
(ma-)(cl6) rafiki (cl9/10) z-angu (cl10) wa-kubwa (cl2) wa-li anguka(a) (cl2) (the "ma" is optional, with it you have a triply mixed concord)
(plural expanse/round swelling(?)) friend(s) (inanimate animal (?)) my big, they in the past fall (My big friends fell) (Rafiki is a loan word from Arabic, which might partially explain its strange classification and behavior)
vi-tabu (cl8) vi-kubwa (cl8) vy-angu (cl8) vi-li-anguk(a) (cl8) (normal concord)
(plural tool/dimunitive) book big my, they in the past fall (My big books fell)
vi-faru (c18) wa-kubwa (cl2) wa-li-anguk(a) (c12) (mixed animate and regular concord)
(plural tool/complex diminutive) rhinocerous big, they in the past fall (The big rhinoceroses fell)
m-penzi (cl1) w-angu (cl1) a-na-ishi (cl1) hapa (normal concord)
(singular human) lover my, he/she now lives here (My friend lives here)
ki-penzi (cl7) ch-angu (cl7) ki-na-ishi hapa (normal concord, despite animacy)
(singular tool/dimunitive) sweetheart my, he/she now lives here (My sweetheart lives here.
Because of the clash between concord based on class and concord sometimes based on animacy, you get a mixed system in this type of Swahili, where a singular word can require different concord with different parts of speech. In each case, the noun itself is said to belong to a single class based on what prefix or lack of prefix it takes in the singular and plural. Apart from this animacy anomaly, class prefixes are quite regular.
