This paper studies a family of M-estimators whose objective functions are formulated as U-processes and may be discontinuous.Notable examples in this family include Hans maxi-mum rank correlation (Han,1987) and Cavanagh and Shermans generalized maximum rank correlation (Cavanagh and Sherman,1998) estimators for monotonic index models,Khan and Tamers rank estimator for semiparametric censored duration models (Khan and Tamer,2007),and Abrevaya and Shins rank estimator for generalized partially linear index models (Abrevaya and Shin,2011).For these estimators,we for the first time obtain Bahadur-type bounds,es-tablishing asymptotic normality (ASN) for all linear contrasts under the "Portnoy Paradigm",namely,the number of parameters,p,in the model is allowed to increase with the sample size,n.The main results show that,often in estimation,as p/n → 0,(p/n)1/2 rate of convergence is obtainable.On the contrary,asymptotic normality requires much stronger scaling requirements than p2/n → 0.Theoretically,our analysis builds on but also extends Spokoiny (2012a) in a sig-nificant way.where the loss function has to be differentiable.For handling non-differentiable loss functions,we establish a new maximal inequality for degenerate U-processes of many covariates,which plays a pivotal role in our analysis and is of independent interest.