We show how a new threshold can be signaled by structure in the transverse momentum distribution of W+/-,Z produced in pp interactions. We construct an explicit model inspired by compositeness where W+/-, Z are produced with increased rates for pT >~ 30 GeV as a result of new interactions with quarks and gluons. Monojet and anomalous (ev-jet) events are obtained with correct rates as a result of Z(-->vv)g and W(-->ev)g production via the new interactions. The transverse momentum distribution of the new events peaks in the 30-50 GeV range and a broad enhancement in (W, Z)-jet invariant mass is obtained in the 150-200 GeV region. We suggest tests relevant to a general scenario of this type.