Until 1984, rent growth and CPI growth pretty much matched each other, meaning inflation adjusted rents stayed nearly constant. And then, a departure began between the two: rents have been rising faster than inflation, and the difference between he two is accelerating.
In a well functioning housing market, rents should stay fairly constant across time. If rents rise above inflation, builders have incentive to build, until they create enough vacancy that real rents fall again (in the old days, this would usually entail a little bit of overshooting).
We do not, alas, have a well functioning housing market in the US. My hypothesis is that this is because builders, unlike, say, auto manufacturers or farmers, need to get permission from governments to respond to demand pressures, and often do not get it.