If you're talking about an entire NBA game, the answer is pretty clearly no, without permission. There are arguments you can make, in terms of you're not looking to make money off the game, but if you just take a 100% copy, that's in all likelihood going to be copyright infringement.
Highlights are a different matter, and taking clips and making them into a mix are different still. There, you can get into arguments related to fair use and the transformation of the original work into something new. I'm not an intellectual property / copyright attorney, so I don't know all the nuances, but I think a mix meant for public distribution, that brings no monetary gain to the creator, would be much more likely to be deemed legal.