It obviously varies state-by-state, but yes most states have a closed season, mostly to allow for either spawning or stocking.

Traditionally (meaning years ago) most states with trout only allowed (trout) fishing from sometime in April to about Labor Day (or even earlier in some states.)

The trend has been for the closed season to be shorter and shorter. Here in Maryland, stocked streams are closed for either one or three weeks prior to "opening day." In Pennsylvania, the trout streams are closed March 1 through mid-April. I believe New York has a more traditional closed season, October or so through April 1.

There are plenty of exceptions to closures; catch-and-release streams are usually open all year, for example.

Bass are regulated slightly differently (at least in MD), you can fish for them, but can't keep them Mar 1 - June 15.

So the answer to your question is yes and no. (Or is that no and yes?) There's always someplace you can fish for something, but it won't be every body of water for every species.