Texas was its own country for a while. It gave up its sovereignty when it joined the Union, but reserved the right to leave. I don't believe that has changed, the Civil War notwithstanding. As such, Texas has considerably different circumstances than the rest of the states.
Texas does not have the right to secede; however, it does retain the right to split itself up into as many as five different states if it so chooses. (see, e.g., this site.)
"It gave up its sovereignty when it joined the Union, but reserved the right to leave..."
If Texas made the attempt, I don't think it would be allowed to leave. It contributes too much to the economy...not to mention all of the military bases located in the state.
Now if Rhode Island, Delaware, or some other insignificant state wanted to secede, I don't think anyone would be bothered by it. lol