Probably not in Full Auto. On the D5100, automatic modes can override user-selected settings, and drive mode is one of the settings that may revert to single-shot. In normal exposure modes like P, A, S, and M, the drive mode should remain where you set it, so continuous shooting is more reliable there.

If you want point-and-shoot convenience with more control, try using P mode instead of Full Auto. It handles most exposure decisions automatically but usually lets your chosen drive mode stay active. Some cameras also offer custom user modes, but based on the answers here, the main takeaway is that Full Auto on the D5100 is designed to force certain settings rather than preserve them.