German is typically described as a Subject-Object-Verb language. For former American President Kennedy's mistake to be grammatical (i.e. without the indefinite article "ein"), why should it not have been: "Ich Berliner bin" (or "Ich ein Berliner bin")?
No, because German has V2 basic constituent order, not SOV.
V2 word order is observed when the the finite verb must be the second constituent of the clause. In many German sentences, the auxiliary verb is the one that is finite, and so it must be in second position. The main meaning-conveying verb appears in a non-finite form, and is flexible with respect to which position it occurs in, and typically occurs in final position. For this reason, German can be mistaken as an SOV language. The German translation of "Peter wants to give the boy the book" is usually rendered as:
Peter wollte dem Jungen das Buch schenken. Peter want.PST the.DAT boy.DAT the.ACC book give.INF
But beyond the requirement that the finite verb be second, the word order is flexible. The following versions are also all acceptable.
Dem Jungen wollte Peter das Buch schenken. Das Buch wollte Peter dem Jungen schenken. Schenken wollte Peter dem Jungen das Buch.
(data is from Wetta 2011)
In the sentence "Ich bin ein Berliner" there is only one verb, which is inflected, and so it must be in second position, after "Ich".
In a true SOV language, either the finite verb must be clause final, or all of the verbs must appear at the end of the clause if there is more than one. Here is an example of a serial verb construction in Barai (Olson 1981, qtd, in Foley 2010), which is a strict verb-final language.
namot tau da-gwe-r witki-r am-ta man sugarcane foc-bite-r tear-r eat-pres ‘The man is biting the sugarcane off and eating it.’
Although languages are often referred to as having V2 word order, it is more correct to say that V2 word order is a property of clausal constructions. Not all clause types in German, for example, have V2 word order; subordinate and question clauses do not, but main declarative clauses do.
Foley, W A 2010, Events and serial verb constructions, Complex Predicates: Cross-Linguistic Perspectives on Event Structure, Cambridge University Press, Cambridge, 1, 79-109
In generative grammar, it has long been argued that German is indeed SOV, that the verb-final order is revealed in subordinate clauses, and that the Verb Second (V2) order in main clauses is due to a complex transformation. The locus classicus for the argument is:
Bach, Emmon, (1962) The order of elements in a transformational grammar of German. Language 38, 263-269. JSTOR
The online syntax textbook by Beatrice Santorini and Tony Kroch has further information about this idea:
@jlovegren's answer is complete, just wanted to add some points.
Another possible reason for German to be mistaken for SOV is that subordinate clauses usually follow a SOV pattern.
Ich weiß, daß ich ein Berliner bin. 'I know that I am a Berliner.'
Usually, in subordinate clauses, the conjugated verb comes finally, whether there is another non-finite verb or not.
Another point to make (after reading @Alenanno's comment) is that V2 doesn't necessarily even look like SOV, since V2 doesn't necessarily mean that the main meaning-conveying verb comes in a final position. For instance, Danish (and other Scandinavian languages AFAIK) are V2, but the main verb usually comes before the direct/indirect object.
Peter ville give drengen bogen. Peter wanted-to give to-the-boy the-book.
If e.g. the indirect object is brought to the head position, the conjugated verb is in 2nd position, but the direct object still comes after the main verb.
Drengen ville Peter give bogen. To-the-boy wanted-to Peter give the-book.