The Washoe, the Northern Paiutes, and the Shoshone of the Great Basin enjoyed gambling. This image shows a group of gamblers in Virginia City, Nevada ca. 1880 (or slightly before). They are using cards, which they adopted after contact. In pre-contact times they used marked sticks.
Like most people, the American Indians of the Great Basin enjoyed storytelling with the traditional dichotomy of fictional stories for entertainment (told in evenings and often restricted to the winter months) as opposed to legends to be believed, which were not for entertainment (but rather for education and to serve as warnings), which could only be told during the daylight hours.
The 11th volume of the Smithsonian Handbook of the North American Indians deals with the Great Basin. I hope that helps.
Some prehistorical Native Americans from the American North-East had games like tokens. Also, some of them had a form of cup and ball game made out of bone. It is essentialy an animal phalanx (like deer) that has been emptied out with a string tied to it. On the other end of the string was a bone fashioned in a pointy way.
The goal of the game would be to swing the phalanx and while it is in the air, the player must position the pointy bone so that the phalanx could land on it.
Of course, like itsallfolklore said, they had other pastimes like storytelling.
Sources:
Roland Tremblay, Les iroquoiens du St-Laurent; peuple du maïs, 2006, 132p.
I know that a very famous story was Fort Michilimackinac being overtaken by a group of native men slowly using a game of bagataway (which is what lacrosse is based on) to suprise the English at the fort and overtake it in 1763. The game itself had fairly developed rules at the time (although not too much like modern day lacrosse) and was used for entertainment purposes.
What do you mean by "entertainment"? Much of the cultural practices of many nations could be called "entertainment" while also serving other purposes. Gambling of course was a big one in a lot of areas. Lahal (the stick gambling game) is still big up and down the Pacific Coast, but gambling over racing in canoes, on horses, foot races, over who could catch the most, well, pretty much anything.
Storytelling and music! Every culture has stories that are told, but here the question of what is entertainment comes in to play. Google gives "the action of providing or being provided with amusement or enjoyment." - This seems to imply some sort of action devoid of practical purpose, or at least references the strong artificial divide in much of Western Civilization places between work and leisure. Not so for most Native American cultures. For example, for many Native American cultures, music was a tool, used for legal purposes (marking agreements), economics, spirituality, health, storytelling, making work go by easier, or entertainment. Teasing out the "entertainment aspect" of music is a rather pointless task, as the people themselves didn't see it that way.
Stories! Most cultures would use stories to pass on history, morality, a worldview, a shared body literature that could be referenced in speech and jokes, also for economic and legal purposes, or to entertain the kids. But again, it's largely pointless to tease those aspects apart. The same could be said of things like Lacrosse, forms or racing and competition, toys, or dolls. All had a purpose and were far more integrated into the culture than might be indicated by the term "entertainment".
Finally, I would argue that most "entertainment" came about in the course of the average day, and was worked in to every tasks and chores in the form of jokes, chatting, competing, and teasing. Doing linguistic documentation, almost all the jokes I'm told are recountings of funny things that happened while working, i.e. while living. Entertainment was a part of life as a whole rather than being a specific subdivision. I suspect that this is the case for us as well, we've just learned to talk about our lives using a vocabulary that leads us to view it differently.