Well, I have seen carousel type tool changers fault out because they don't see the holder in the socket it rides in. It's kind of like the program calling for a tool that doesn't exist. Tool 21 in a 20 tool changer, for instance. It happens when a program from a large tool changer is sent to a smaller machine without edits to make do with the tools available. Or if there is a typo in the program calling an empty socket in the changer. A chain type tool system can have the same type faults. In the case of a program calling for more tools than the system will handle, there is a different code for "hand load" a toolholder. Seen that happen when going from a large machine to a small machine in a pinch. Or to one without an ATC. All hand load. Have to be careful on those to load the right tool.
When you say the tool stuck and blew air, I am guessing you mean the spindle. Most have a drawbar that pulls a retention knob up into the spindle taper. It's important to keep the tapers clean The air blast is a little help to clear loose chips if any manage to get up in there, but the toolholder has to drop out before that does any good anyway. I'd be suspicious of the toolholder that stuck. Look for any dings or debris. Rub it over with ScotchBrite (the taper), and clean out the spindle. If this is an older machine, the gripper dogs on the drawbar may not be releasing properly. That generally calls for a replacement of some part of the power drawbar. It's a pneumatic cylinder up on top of the spindle housing. Sometimes a sharp rap with a brass or copper mallet sideways on the toolholder will release a stuck toolholder. Just be ready to grab the holder if it falls out. No need to ding the table, part, your hand, or break the tool in the holder.