I have a pet peeve that I really must get off my chest. Yes, another one! Rest assured, there are plenty more where this comes from.
Logit and Probit models are the joint work-horses of a lot of microeconometric studies. They've been widely used for a very long time, and these days they are frequently used with what I'd call pretty large data-sets - ones where the sample size is in the thousands, or even hundreds of thousands.
These models are invariably estimated by Maximum Likelihood (ML), and as we all know, ML estimators have great large-sample (asymptotic) properties. Specifically, they're weakly consistent, asymptotically efficient, and asymptotically normal - under specific conditions. These conditions are what we usually term the 'regularity conditions', and these are simply some mild conditions on the derivatives of the underlying density function for the random data in the model. The normal and logistic distributions satisfy these conditions, which were first introduced in the context of the formal properties of ML estimators by Dugué (1937) and Cramér (1946). So, what could go wrong?