Sure. My first iMac was a G3 and it overheated. My second one was a G4 and it overheated as well. If yours ever starts shutting down for what appears to be no reason, back up all data immediately. The CPU is shutting down the computer as a safety precaution because of overheating. There are free programs for the Mac that monitor your temperature.
I have a G4 Power Mac that is 8-years-old and is duel processor. Running OSX has made it stable. The only big problem I had was a harddrive that died on me and needed to be replaced. Otherwise doing fine.
The G4 Mac Cube was designed without a cooling fan and worked fine without them.
There were some problems with heat on some first edition laptops in both the G3 and G4 but it wasn't generally related to any crashing... just hot laps. i did not experience this eitherprimarily because I didn't purchase any of those first edition laptops.