The only state that can split itself up is Texas, which was part of the agreement to bring the Republic of Texas into the union. http://en.wikipedia.org/wiki/Texas_divisionism
It requires both State action and federal legislation. But the State action is clearly a required step.
> The only state that can split itself up is Texas, which was part of the agreement to bring the Republic of Texas into the union.
As your own link notes, that agreement specifically cited that it could only be done subject to the requirements of the Constitution, which would seem to mean that Texas can do it exactly as much as any other state can -- that is, with action of the State legislature and the federal Congress.
As your link indicates, a more accurate telling is "The only state that some proponents argue can split itself up is Texas".
Puerto Rico already voted in favor of becoming a state in 2012. The only thing stopping it from happening now is a simple majority vote of Congress[0].
Personally, I think giving Puerto Rico, a territory that already must obey federal laws, the ability to vote on said federal laws is a much higher priority than splitting up the states that we already have.
...which is to say, of course, that I imagine that none of these will happen anytime soon.
[0] As best as I could tell, the threshold for statehood really is this low.
The Civil War pretty much screwed this up and the division can happen elsewhere because of precedent.