Ime it depends on species and circumstances. I've seen it happen both ways, fins grow back normally or never quite do, both when the spines themselves have been damaged or cut off. Most recent example where they did was with a C. gibberosa kapampa I have that was in a tank at my wife's business where a fire demolished the building. At some risk, I was able to get in there during a lull in the fire to retrieve some property, including the fish from this tank. The water in the tank was dark brown, with the glass lids broken, debris fallen into the tank, and the kapampa's fins damaged, one of the pelvic fins being just a nub. Long story short, after several months not much happened and by that time I figured that was that for the bad fin, but later it began to grow discernibly, much to my surprise. By about two years it was 75% regrown, now it's length is about 90% of the other undamaged pelvic fin-- these are long fins on mature Cyphotilapia. My supposition is that the initial period of not much discernible happening was spent repairing the base of the fin and its rays before any visible growth could take place.
This was not a young fish, it's 10.5 years old right now and was 7.5 at the time of the fire.
Again, like I say I've seen it go both ways-- with the rays themselves being nipped off or damaged. Considering that some animals can regenerate whole tails or limbs, it's not like it's a biological impossibility. Don't know what I did with the reference, but used to have a link to a science paper that delved into the complexity and bio-chemistry involved with fin regeneration. I don't remember the details, but, basically, certain bio-chemical events have to happen in the right sequence and there are also environmental factors. It's not a simple process, there's some sophisticated science to it.
Simplistically, my own experience is sometimes it happens and sometimes not. Specifically regarding dorsal fins, the case I best remember was a betta I had many years ago. In an apparent close call, I found it one morning with nearly all its fins chomped nearly clean off by a larger fish (which I subsequently moved). All fins, including dorsal eventually grew back to normal. In that case the process started within a few weeks or so, the fins initially growing clear. Whole process took a few months, and eventually the replacement fins were fully colored, though a little different shade than the originals... strange as it may sound, I remember that fish well, called him Rambo after that.