NASHVILLE, Tenn.—Lydia Harrell worries it is only a matter of time before she gets priced out of the red-hot apartment market here. Since she moved to the heart of the country-music scene two years ago, her monthly rent has risen 22% from $750 to just over $900.

A few months ago, she and her roommate considered moving to a different building, but they discovered that similar downtown apartments were renting for well over $1,000. "It's starting to get a little crazy. If it gets any higher we'll have to go somewhere else,"...