There are brown recluse spiders where I live. Is it possible to get a bite and not feel anything?

Possible. It is possible to be bitten during your sleep and not feel anything. However if you do not have any signs or symptoms of a recluse bite then you should not worry about it.
Yes. Brown recluse spider's bite can cause no reaction, a mild reaction, an immediate or a delayed reaction. With a significant reaction a which blister appears after bit. The surrounding skin becomes hard, swollen & can be red, white & blue. A deep lesion can develop which can eat through fat and muscle.