If you chose a $2^{15-11}$ resolution $\mathrm{III}$ design (e.g. this design from NIST) then you will have some main effects completely confounded with two-factor interactions. This means that the run levels of some interaction effect $AB$ will be identical to some other main effect $G$. You will not be able to fit a model with both $G$ and $AB$ using the standard analysis techniques.
Consider the interaction $AB$ again. There's no reason why the main effects $A$ and $B$ would be significant if $AB$ is significant. Each effect $A$, $B$, and $AB$ describe something different. There's a principle called heredity (or hierarchy, or another name that escapes me) that says that an interaction should only exist if at least one (or both) of the main effects is significant, but this is a sort of rule of thumb.
When fitting resolution $\mathrm{III}$ designs sometimes people use the heredity principal and look at the active main effects and which two-factor interactions are confounded with them. If $A$, $B$, and $G$ are significant and $G$ is confounded with $AB$ then the model with $A$,$B$, and $AB$ active but $G$ inactive may be plausible. Usually a followup experiment is conducted to disambiguate the situation.
This is all covered very thoroughly and with better examples in Design and Analysis of Experiments by Montgomery (2012). I'd suggest reviewing this text.
