  We consider a class of defocusing energy-supercritical nonlinear
Schr\"odinger equations in four space dimensions. Following a
concentration-compactness approach, we show that for $1<s_c<3/2$, any solution
that remains bounded in the critical Sobolev space $\dot{H}_x^{s_c}(\R^4)$ must
be global and scatter. Key ingredients in the proof include a long-time
Strichartz estimate and a frequency-localized interaction Morawetz inequality.
