I've heard many times throughout my life that, when they became a state, Texas secured the right to secede whenever they liked. I never had any particular reason to question it. I mean, it was always a kind of nice-sounding idea; one less state with massive support of guns and Republicans, more progress for the rest of us on the federal level. It turns out, though, that that's just another element of puffery perpetuated, presumably, by Texans who want to feel special.
They do, apparently, have the right to split themselves up into 5 states within the US, without Congress's permission. This would mean that Texas's total population would have five times as much representation in the Senate, though I believe that it wouldn't affect their representation in the House of Representatives.
After the 2008 election, recent mathematical superstar Nate Silver wrote up a theoretical breakdown of Texas into five sub-states, linked here, via this article on Slate. That article also addresses Texas's only real shot at secession:
Could the current crop of Texas secessionists use the division clause in pursuit of their separatist goals? It would certainly be worth a shot. Naturally, it took the Machiavellian political mind of Texan Tom DeLay—the former House majority leader, currently out on bail while appealing a 2011 money-laundering conviction—to put the pieces of a tenable scheme together. The day after Perry blew his secessionist dog whistle to that reporter back in 2009, DeLay went on MSNBC's Hardball to cheerfully defend his governor's remarks. When host Chris Matthews insisted (correctly) that unilateral secession was illegal and couldn't take place, DeLay stopped his maniacal grinning for a moment and cited the division clause.
In a sign of just how much the two political parties' fortunes have shifted in Texas since the days when John Nance Garner represented the state in Congress, DeLay intimated that the threat of sending eight newly minted, and almost certainly Republican, senators to Washington might be the key to getting this whole secession ball rolling. Referring directly to the language of the joint resolution, he said, "If we invoke it, the United States Senate would kick us out ... because they're not going to allow 10 (sic) new Texas senators into the Senate. That's how you secede."