I've read in several places that it is reasonable to assume that the usual conditions (that the filtered space is complete, and that the filtration is right-continuous) hold since one can always minimally modify the setting one is working in so that they do hold (simply complete the space and replace the filtration with its right limit).
I've also been told that modellers can find doing the above unnatural since, for example, there is no reason the information available to us at time zero should include that about events that happen in the future, even if they have zero probability of occurring. More specifically, suppose that $X$ is a real-valued process on some filtered space $(\Omega,\cal{F},\cal{F}_t,\mathbb{P})$, $B$ is some Borel set and that $\mathbb{P}[X_T\in B]=0$ for some fixed $T>0$. Then it seems a bit bizarre to assume that $\{X_T\in B\}\in \cal{F}_0$, that is, that at time zero we know whether or not $\{X_T\in B\}$ has happened.
However, I don't know enough of the big picture to grasp the real consequences and potential issues of assuming that the usual conditions hold. Could someone explain them or a few of them? I'm not talking about the technical advantages of assuming that they do hold (I'm thinking of asking a second separate question about this), but more something like "can we ever find ourselves in a situation we where assuming they do hold is problematic and there is no easy way around the issue?".