It depends.
Many casino cages will cash a limited number of low-denomination chips from other properties, especially if the casinos fall under the same corporate umbrella (Station Casinos, MGM Resorts International, Caesars Entertainment, etc.), but they don't like to.
There are practical reasons behind casinos' reluctance to accept "foreign" chips. For starters, ultimately all chips need to be cashed at the casino that issued them, so if you don't do it, the casino to which you hand them over will have to take care of getting them back to their home casino. This means physically sending someone out on a periodic "chip run," visiting every casino whose chips their property has acquired -- and that requires time and money.
Casinos also like to keep their own chips on property in order to keep track of what their players are up to. For example, after a successful night, a favorite ploy of gambling teams (whether honest professionals or cheating crews) is to divvy up the spoils and send the team members out to different properties to cash them in. That way the casino they've hit can't be sure how much was won and by whom -- exactly, of course, what management wishes to avoid.
Although casinos are fiercely competitive with one another with regard to taking your money, they also tend to stand together when it comes to protecting themselves from undesirable players. Having a players card from the casino that issued the chips will sometimes help to smooth the process; it shows you're known to that property. But if a player presents chips either of a high denomination (over $25) or that total a large cash value, any cage will almost certainly run a Central Credit check to see if there's a "hold" on them for any reason.
Why would there be a hold on chips? For one, a casino might suspect that its chips were acquired in a dubious manner. For another, the player might have an outstanding marker at the casino that issued them and is trying to cash out, at another property, what's left in his pocket, rather than putting it toward settling his debt. If for any reason the chips presented are shown to be on hold, they will not be accepted.
What if you want to play your chips rather than cash them in? Usually, before you can sit down at a table, you have to exchange them for house chips at the cage.
Those are some general points to bear in mind, but specific policies vary from property to property. Since there are so many potential restrictions, our advice is to cash your chips in when and where you win them.
All that said, the bigger the player you are, or appear to be, the more accommodating any casino will be. If it's a choice between accepting chips from another establishment and seeing you walk out the door, most places would prefer you to stay and might make an exception.