Regression analysis with small but complete dataset (fully representing reality)?

I have a repeated measures design with about 16 cases and 5-6 points of 
measuring. Sometimes, 1-4 full cases or some points of measure are 
missing. (The measures are 20 numerical and categorical data taken from 
questionnaires.)

The clue is: It's a small dataset with holes in it, but the 16 cases are 
all that even exist. So they fully represent reality wherever they're 
complete.

I wanted to run logistic regressions with up to 6 predictors. But can I 
do that? I know about the many problems such small datasets have for 
regression analysis - but do they matter as much if there aren't any 
more cases in reality?
Are descriptive analyses the only ones I can use?

Many thanks