For example, a very widely used process for modelling information-driven timeseries (like stock returns) is the jump-diffusion model where the diffusion component is a Brownian motion while the jump component is a Poisson process.
The underlying volatility is often modelled using a different process - e.g. the SABR model, the Heston model etc.
There are similar cases for interest-rate processes (Hull-White/BDT etc) which have to satisfy conditions of mean-reversion and no-arbitrage across the yield curve.
See, it's not as if we don't realize that the underlying processes are mathematically inadequate to fully explain all market movements. But for a model to be useful, it has to satisfy two conditions:
1. Be able to produce a non-arbitrageable "mid"-price for making a market (ie if someone asks a trader to quote the bid-ask for an option, e.g.)
2. Be able to reproduce the current market prices of an asset/its derivatives. This requires model calibration.
Models are chosen based on how easily and how fast they can satisfy 1 &2.
Do they produce risk numbers that are believable? Probably yes, if the model has been calibrated and has been tested against out-of-sample inputs.
Do they guard adequately against event-risk (the thing you would try to signify using your "infinite" variance distributions)? Probably not - but then again, nothing does. How would you go ahead and calibrate the Levy distribution so that the sampling process can explain currently tradeable market prices? Would be it a "do once, leave forever" calibration? Or would it change from day-to-day (ie "local" calibration)?