Yes. I was diverted two days in a row on an ATL-ROA flight last Christmas... once to PIT, once to TRI. It was such a comedy of errors I was living a pretty good approximation of a Lifetime Christmas movie! In both cases, a proportion of the passengers chose to deplane and rent a car instead (or simply remain in the city) - and Delta removed their baggage upon request, time permitting.
In my experience, you usually have the option to deplane if you're diverted and the cabin door is opened at a gate. If it's a very brief stop to wait for weather to clear and the pilot may need to take off at a moment's notice, though, they are less likely to offer the option. Other situations making deplaning unlikely are arriving at an airport that isn't usually a DL station, or being diverted on an international flight.