It's pretty rare for people in their twenties, but more common amongst older generations.
Usually you're forced to get a landline if you want DSL. I've never seen a cable company force a landline on you. You can also get around the landline restriction from the phone company if you get a fiber connection though that's not offered in all areas.
The only reason why I have a landline (I'm 33) is because I have kids. In an emergency, the last thing you need to worry about is whether or not you can get a cell signal at home.
That being said, we just moved and my wife just reminded me that neither of us actually know what the new number is - we've lived here for almost a month. So, we still rely on our cell phones.
I personally wouldn't have a phone at all if it weren't for my job requiring a personal cell phone (cheap-asses won't provide a company cell phone) -- and I have 2 kids. The whole "emergency" thing never sat well with me as a reason for a phone or to carry a cell phone.
We had a really bad storm a few years back when my first kid was only about 3-4 months old. That house was at the low-point for the block, so water would pool in our backyard when there was a lot of rain (and the city has horrible storm sewers). Our house has a nice basement with a great drainage system that keeps it dry - so long as the sump pump can run. The rain was horrible: ~5-6 inches overnight. There was also a tornado warning, so we were in the basement for most of the storm.
Then the power went out.
Now, I had installed a battery backup sump-pump six months earlier, so that was able to keep up for a while. Our cell service also went in and out as the power outage was very large and our service at home was spotty (and the towers were over loaded). The only way for us to get ahold of family for help was through the land-line. I still keep at least one plugged in phone (non-cordless), just for this reason. Because of the land-line I was able to get my father-in-law to bring over a generator to power my sump-pump for the next day or so until our power was restored. Our basement had almost no water damage, mainly because we kept a land-line.
I have Comcast and they basically force you to get a "land line" through them (although I guess it's closer to VOIP?) for a cheaper package. Mine isn't plugged into a phone though, I don't even know what the phone number is.
Usually you're forced to get a landline if you want DSL. I've never seen a cable company force a landline on you. You can also get around the landline restriction from the phone company if you get a fiber connection though that's not offered in all areas.