Without data or detailed outputs, it is a tough question to answer but the first thing that comes into my mind in such a situation is that there is a low number of observations, at least for specific behaviours. For example, if you have thousands of individuals but only a few dozens that are relevant to calculate a coefficient in a regression, one may expect the S.E. of such a coefficient to be quite important.
Moreover, in case the residuals would not be normal, asymptotic formulas would not be valid anymore, so in addition to have a high standard error, these interval would not make any sense anymore.
If you want more comments/suggestions, I guess you will have to provide more details about these specific situations where "the S.E. become very large or NAN". R codes are welcome.
Are you trying to do logistic regression? If so, Very large SE's can appear when there are perfect predictors. This occurs because of a division-by-zero error; the information matrix blows up.