Yes, Sumerian scribes did sometimes write words entirely or partly phonetically using syllable signs. This could occur for several reasons:
As Draconis already noted, grammatical prefixes and suffixes (which Sumerian used a lot, especially with verbs) were always written phonetically, since that was really the only way they could be written. (Admittedly, sometimes they might just be fully or partially omitted, if the grammatical role of the word was clear from context.)
Also, some words were always written with syllabic signs simply because they had no logographic sign. One example that comes to mind, off the top of my head, is ummia = "expert, teacher, master" which, as the linked ePSD page shows, is invariably written as either um-mi-a or um-me-a. Loanwords from Akkadian to Sumerian (which ummia might also be; the Akkadian ummānu seems clearly related, but I'm not sure which way the loan may have gone) are also fairly often written this way.
Sometimes, even if a word did have a logographic spelling, it could also be written syllabically e.g. because the logographic sign was obscure or potentially ambiguous. There are quite a few such words; one that comes to mind is ašag = "field", which could be written syllabically as a-ša3, logographically as ašag = GAN2 — a sign that could also be read as the more or less synonymous word gana = "field", or as the unit of measurement iku = "field (unit)" — or even by combining both forms as a-ša3ašag, where the logogram is accompanied by a phonetic complement to indicate the intended reading.
One special case worth mentioning is Emesal, a dialect / sociolect / special register of Sumerian whose origins are obscure, but which appears to have been used as a stereotypically feminine mode of speech in various literary texts and cultic songs. (For more background, see e.g. Whittaker (2002).)
Emesal is all but invariably written syllabically, in a sort of "eye dialect", to highlight the differences in pronunciation from standard Sumerian (Emeĝir), even when such differences may not actually be apparent from the spelling to a modern reader. For example, the word dumu = "child" is written in Emesal as du5-mu — a spelling that was presumably read in some way differently from the standard Emeĝir reading of the logogram dumu, but if so, we don't really know for sure how.
The Emesal forms of other words (of which ePSD has a nice list) could be more clearly distinct from the Emeĝir pronunciation; for example, the Emesal form of the word diĝir = "god, deity" could be written as dim3-me-er, dim3-me8-er, dim3-mi-ir or di-me2-er, presumably reflecting a pronunciation like dimmer or dimmir. In some cases the Emesal forms could even be completely different words, as with the word nin = "lady, mistress" whose Emesal counterpart (commonly encountered as a title of female deities) was gašan (most commonly spelled ga-ša-an).
One particular context in which syllabic spellings are found is in lexical lists where they were deliberately used to indicate the (approximate) pronunciation of words. Such lists are one of the main reasons why we know anything about the pronunciation of Sumerian logograms at all. However, it should be noted that most of these lists were compiled by Akkadian scribes, as tools for teaching Sumerian after it had become a mostly dead language, and thus the pronunciations recorded in them may be filtered through Akkadian phonology.
Finally, it should be noted that the line between syllabic and logographic spellings in Sumerian can sometimes be kind of fuzzy, because Sumerian has a lot of monosyllabic words, and even many of the polysyllabic ones are actually compounds of identifiable monosyllabic elements. And, on the other hand, cuneiform signs can also be combined in various ways, and it's not always too clear what should be counted as a single sign.
As a familiar example, the word lugal = "king" is etymologically a compound of lu = "man" (written with the sign lu2) and gal = "big", and the sign most commonly used the write it (usually transcribed as a single logogram) is a compound of the corresponding signs — although written in the reverse order (lugal = GAL+LU2) for some reason. As ePSD notes, the spelling lu2-gal, with the signs in the opposite order, is also attested in a handful of places; it could be considered a phonetic spelling, or just a variant of the compound logogram.
As another example, consider the sign agar3 = LAGAB×A+GAR, one of the several ways of writing the word agar = "meadow". Looking at the sign, you can see that it's basically the phonetic spelling a-gar with a square drawn around it. Such compound signs with other signs drawn inside them are quite common in Sumerian cuneiform, although they tend to become less common in later forms of the writing system (with the enclosed signs either being moved outside the enclosure or being otherwise simplified).