There's 3 states your hero can be in regarding this. Normal, tired and exhausted. Normal means no stamina drain. (in fact, a slow stamina recovery over time though minimal) The status ticks over to tired if you spend about a day without sufficient rest. That means minor stamina drain as time goes on. When enough time passes without sufficient rest, the status becomes exhausted meaning you get heavy stamina drain. This won't kill you as long as you keep your stamina above 0 (otherwise health will go down), but the only way to cure it or prevent the tired/exhausted status is to get about 8 game hours of sleep a day. The message you get is telling...a message about "awakening much refreshed" or "fitful sleep" means you slept long enough, a shorter message means you didn't get enough sleep. You'll regain hp/sp/mp either way, but only the long naps cure or prevent the tired/exhausted status.
This is all very much intentional.
