Most definitely, my apartment before buying a home was ~$100/month less than my current mortgage, now (3 years later) the rent for the exact same apartment is ~$300/month more than my mortgage.
Isn't that counting inflation twice?
California's recent rent control laws caps rent increases at 10% or 5% + the yearly CPI increase (whichever is less) every year, for example, explicitly allowing rent increases to outpace the general inflation rate. Personally, the rent I pay has gone up 9% this year.
Rents can change based on whether an area becomes more or less desirable, as insurance or other costs for landlords change, as ownership of rental units consolidates, etc.--that is, for many reasons that might be only tangentially connected or even completely divorced from the reasons that the costs of other goods and services are changing.
One thing that I think this actually shows pretty well is the notion that part of the answer depends on how long you intend to stay in a place. If you expect to move in two or three years, it's probably not worth the hassle/expense of buying. If you intend to stay 5+ years in one place, it makes more sense to buy.