Were I to give an exception, it would be that perhaps one or two other nations could have formed and coexisted with the OTL countries ITTL.
Sure, but it'd be tricky, especially if you don't want one of the new countries to be a puppet state.
One of the easiest ways is for a relative Mexico-wank TL, where Mexico has more control over its northern California territories. Then the Mormons in Utah could potentially play the US and Mexico against each other as a buffer state,
independent despite being insular. However, should Mexico and the US ever decide that it would be better to wipe off the abomination of Mormonism (tm), Deseret would be gone.
Another possible nation (and an indian one to boot) might be the Iroquois Confederacy. While it was split by the ARW, intact it was one of the strongest and most modern indian federations around. If rather than split all the tribes decide to support the US in exchange for territorial integrity, they could keep modernizing until the point where they are caught up to the US and Canada, at which point they can balance the US and Britain against each other.
However, in my opinion Texas makes a horrible independent nation. Not only is it far too big and empty (how does one keep an indefensible border?), as well as poor and scarcely populated, but under any condition it would be bordered by multiple expansionist nations who it could not defend against. Texas was broke defending against Mexican raids, let alone a real land grab. Any independent Texas would see itself nibbled away bit by bit.