Well, yes and no. I'd say there is a whole bunch of you in the multiverse, having all sorts of experiences simultaneously. I would say that your experiences are macroscopic and the boundaries between them, between you-here-now and another-you-there-then are kind of fuzzy.
At every moment, all sorts of quantum superpositions get decohered around you one way or the other. Say if a photon just landed on your forehead, it won't matter that much to you whether it was horizontally or vertically polarized, your experiences will not be affected and you-here-now branch will include both alternatives. On the other hand if a stray cosmic ray hit a cell in a DRAM chip and crashed your computer, one of you would never read this message so you-here-now branch would split and diverge at that point. But between these two extremes there would be a gray area where it would be very hard to tell whether your experiences are sufficiently different to count it as a split.
Yes, I agree, this is a very good question to ask. I also admit that current answer is not entirely satisfactory: some people dismiss it by saying since it is the same old formalism it produces the same answers and doesn't require a separate proof, other people say they have proved it and yet other people say that all those proofs rely on circular arguments and are therefore invalid. I tried to follow these arguments and got seriously bogged down, so I don't have an opinion one way or the other but my gut feeling is that such proof should be possible.