Yes, it also takes into account rising quality. For an example, in 2010 I rented a rat hole apartment for $x from a fisherman who had inherited the building. He never did maintenance (he was out to sea most of the year) and he never raised rent.

A large company bought the building after I moved out. Ten years later, the same apartment with a fresh coat of paint and new countertops was back on the market for a rent of about three times $x.

The CPI can say that apartment, since it was refurbished, increased in quality and so it wasn't really a price increase of the same good from $x to $3x. This offers a "degree of freedom" to adjust the CPI itself (since quality is inherently subjective), and may be a big part of why CPI does not reflect the lived experience.

I didn't care one bit about paint or countertops when I rented that apartment and I assume broke young adults today don't either. At the time I wanted the cheapest place to live in the area and this was it. It still is one of the cheapest places, but you need three times as much money to rent it.