There are a number of situations where you'd want to remove a wire from its connector. Whether or not you can do it yourself depends on the type of connector. Some require special extractor tools, but many others don't. Most computer fans have connector pins that a removable with only a paperclip. It's easy once you see how to do it.
This photo we found on the internet shows how it works very clearly. The little blue arrow shows the "tang" or "barb" that holds the pin in place in the connector housing. You gently press down the tang with the end of a paperclip while simultaneously pulling on the wire and the wire will slide out. Do it gently so you don't crush the tang! When you push the pin back into the housing, the tang will pop into place and hold the wire in place, a lot like a fish hook.
The series of photos starting with "B" shows another common connector style. But here, there isn't a tang on the pin holding it in place. The locking mechanism is on the plastic housing. You can pry up the lock very, very slightly with a razor blade or thin screwdriver and slide the wire back. DON'T pry the lock up too much or it won't lock properly afterwards.