Are highly nitrated cyclopentadiene compounds possible?
Mono-nitrated cyclopentadienes and their benzo-analogues (indene, fluorene) have been prepared in the context of studies on nitro-aci-tautomerism.

As far as higher substitution is concerned, there seems to be at least a crystal structure of potassium dinitrotrichlorocyclopentadienide reported in Acta Cryst., B

In the abstract, the authors (Y. Otaka, F. Marumo, and Y. Saito) mention the preparation of this compound by Haga in 1969, (sorry, reference is behind paywall) via 


  nitration of hexachlorocyclopentadiene with alkali nitrite in an aqueous or ethanolic solution


NOTE: Hexachlorocyclopentadiene once was a favoured starting material in the synthesis of insecticides via Diels-Alder reaction. However, it is a potent teratogen and the "mother" of least five members of the dirty dozen (persistent organic pollutants) nowadays banned for good reason.

You might want to read the MSDS and then decide that you'd rather prefer to work with something else.