Yes, but not by the same person (usually).

Think of it like evolution. Every parent that has ever lived had offspring that were the same species as it, and every one of the offspring were a single species, rather than multiple different species simultaneously. Nonetheless, groups of offspring (populations) may drift apart gradually, as they accumulate increasingly different sets of genes.

The case for language isn't quite so clear cut, because a person's pronunciation of the same word will fluctuate over the course of a lifetime, or even at the same time in different contexts (e.g. saying going at school but goin' at home). But, in general, your phonemic inventory, your linguistic genes as it were, will be highly similar to that of your parents or home environment, but over time, as populations are separated and isolated, differences will be accumulated.

To go back to your question directly, let's take a historical example in English: the pronunciation of the letter R. Today, in most forms of British English, syllable-final R's are not pronounced, while in American they are; dialects of English are thus classified as being either rhotic (with final R's) or non-rhotic (without). Originally, however, all dialects of English everywhere pronounced the R (otherwise, why would we all write them?); around the 18th century, however, speakers in southern England began to drop the R's, and it spread from there. But it didn't spread everywhere: dropping R's didn't catch on in most varieties of American English, though it did in some, and there are even some British dialects that retain the original pronunciation.

So, to answer your question "Was there a point when two versions of a word would be spoken at the same time," the answer is absolutely, because we're living at such a point right now. Languages are in constant flux, and gradually accumulating those small differences.

Likewise, there's a current shift going on in the US called the Northern Cities Vowel Shift. It's common in some areas but has yet to catch on among most speakers. So, even among current speakers of American English, pronunciations of words differ.

This seems more like a question for /r/linguistics.

Could you clarify what you mean? Are you talking about a full-scale transition from one language to another, or a sound shift within a language?