Depends upon the stability characteristics of the aeroplane - but yes, that's common to quite a lot of GA types. Some won't do that however.
As an occasional instructor I've seen inadvertent spiral dive entries quite a few times - two entry mechanisms. (And it's worth mentioning that I'm a CRI, so this is all in people who already hold PPLs).
(1) Steep turns, failing to keep back stick pressure, or keep the ball in the middle.
(2) Failure to properly co-ordinate a turn, especially at low level where there's a changing visual horizon.
I've had a post-PPL student put an aircraft into a spiral dive whilst initiating a descending turn onto deadside during an overhead join. That was exciting!
Loss of horizon in poor visibility, but legal VMC, is another possibility.
G