Me went downtown.
That title sounds like a media stereotype of Jamaican English - even though none of the Jamaicans I've ever met talk like that. In any event, it's not intended as a fictionalized representation of any actual English utterance. It is an example of something that most English speakers can't (or at least generally don't) say.
Why is it that English speakers regularly say things like "Me and Tommy went downtown," but do not say "Me went downtown"?
There are a number of possible explanations for this observation. Let me roughly divide them into two camps.
"Syntactic" sorts of arguments would relate this to some formal rule in the syntax of English. A likely explanation would relate to case. That is, there is (according to linguistic theory) a rule of syntax that assigns case to the nouns in a sentence: subjects are nominative1, direct objects are accusative, and so on. When the subject of the verb phrase went downtown is a pronoun, only a nominative pronoun (I, you, she, he, it, we, or they) will do.
But, unlike many of the world's languages, modern English does not mark case on nouns. Tommy can be either the subject or the object of a sentence. Various theories suggest that when the subject is a complex noun phrase including a full noun like Tommy, the case-assignment function is somehow blocked from getting at the pronoun. How does this work? Formalist linguists have many arguments about this sort of thing.
Over in the other camp, "emergence" or "frequency" sorts of arguments might suggest that the appearance of "me and Tommy" in other contexts, such as the object of a sentence (Mom loves me and Tommy) or non-sentence utterances(Who will chop the fire wood? Me and Tommy.), cements it in speakers' memories. When we construct the sentence about going downtown, we call up this chunk from memory.
Why, then, don't we call up me and then construct "Me went downtown"? There are a number of arguments that could be made. For instance, the memory may store bigrams or n-grams (chunks of n-number of words). A speaker has likely heard both me and Tommy and Tommy went, so "Me and Tommy went" is no great trick. But that speaker is unlikely to have heard me went (in most dialects of English). Cognitive scientists, including (less-formalist) linguists, psychologists, and others have many arguments about how this sort of thing might work.
This two-camp division is, of course, rather simplistic, and there are other sorts of arguments that I'm neglecting. The upshot, though, is that "me and Tommy" behaves differently from "me", whether by rule or by practice.
1. These are somewhat "classical" terms for the cases, perhaps better suited to Old English (which had five cases) or Latin (which had six), than to Modern English (which, depending on how you count them, probably only has three). The Modern English cases are often called subjective (basically equal to nominative), objective (which relates to dative, accusative, and instrumental in the classic nomenclature), and possessive (roughly equal to genitive). Since the subjective/objective divide is only commonly seen in pronouns, some sources (including The Oxford Companion to the English Language) even collapse these into common case.
[Hat tip to Robert Lawless]