In a recent court case from the U.S. Bankruptcy Court for the Southern District of Texas, Judge David R. Jones ruled that Caesar's Palace and another establishment known as Cosmopolitan of Las Vegas, have valid gambling debts that they can enforce in Texas. Nevada Property 1 LLC, et al. v. D'Amico, Adv. No. 13-03041, U.S. Bankruptcy Court, S.D. Texas 2013.
The gambling houses had brought suit in bankruptcy court, trying to have their gambling debts declared not to be discharged in the personal bankruptcy filing of one of their former customers. They were trying to recover over $1,125,000 in "markers" or credit that they had extended to the gambling patron during what was apparently a very unlucky gambling soiree in 2012.
The Court ruled that Nevada law applied, and that the gambling establishments could enforce their debts in Texas. However, Judge Jones ruled that the companies did not make out a case for "willful and malicious injury" under 11 U.S.C. Sec. 523(a)(6), such that the debts would not be discharged.
The Court also ruled that Cosmopolitan and Caesars failed to meet their burden of proof that the debtor defrauded them under 11 U.S.C. Sec. 523(a)(2)(A).
So, so long as you did not intentionally cause the gambling losses (why would anyone do that?) and you did not defraud the casinos (trick them into thinking that you had the money when you didn't), gambling debts should be able to be discharged in bankruptcy.
However, watch out for possible criminal liability. At least in Nevada, I understand it can be a felony to not pay a gambling debt, and I have heard of Texans being arrested for felony warrants out of Nevada for not paying gambling markers.