Well, first of all, it is a bit difficult to explain why something didn't happen, especially when talking about countries breaking up, since it happens very infrequently. It is kind of like asking, "why didn't United Kingdom break up after the end of the British Empire". Or, "why didn't Indonesia break up after the end of Suharto rule"?

But to answer your question - the regional authorities in Russian never had sufficient power or authority to make something like this to happen. The Soviet Union dissolved in part because, at least legally, the constituent republics had constitutional rights to secede. This right of secession was evoked by Russia itself - and since Russia was the largest component of the USSR, that effectively made its continued existence impossible. No such thing happened or could happen in modern Russia. There was no right of secession, and there was no large or powerful enough block or region that could effectively "break up" Russia by seceding.

Also there wasn't really much support for such a breakup. There were a handful of separatist movements, to be sure, but they were relatively small and limited to the Caucasus region of Russia. Central authorities in Moscow made quite a bit of effort to put down this separatism. However even if this movement were successful, it would still hardly qualify as a "break up", merely a loss of a couple of tiny provinces.