Can I add more cases and/or predictors to existing set of data? If I have a set of data with 40 cases and 3 predictors can I add more cases and/or predictors later to the existing set of cases if I want to explore the effect of more possible predictors? The data are calculated from experiments and I want to avoid doing extra work.
 A: It's not very clear what you are driving at here, as exploring the effects of other predictors necessarily entails adding them to your dataset. 
Without more information, 40 cases seems a distinctly small dataset for analyses such as multiple regression (implied by your tag) with 3 predictors. There is no magic threshold that makes analyses convincing or unconvincing, sound or unsound, as that depends on the patterns in the data, but I suspect wording such as "that's a rather small dataset for what you propose" will come naturally to anyone with experience in data analysis. (It is a rather small dataset for any purpose.) Loosely, adding more predictors may hinder as much as it may help. (At the bottom end, the number of parameters being estimated puts a limit on minimum sample size, but that won't bite you according to the information you give.) 
If you are hoping, or expecting, that statistical methods are a way of adjusting for or compensating for a small dataset, beware that sometimes statistical results often indicate that your sample size is too small to say much reliably. Historically, a major role of significance testing is to stop you making a fool of yourself by over-interpreting results from small samples. 
This answer is likely to seem too general or cryptic, but I am puzzled about the underlying question here. 
Substantively, it may well be that in your field, measurement is very expensive in terms of time or effort, and that's to be considered understood. 
