A big problem with jetpacks is that human ankles are weak landing gear. You can't do a parachute landing fall while wearing a jetpack; you have to do a standing landing. With all the mass of the gear on your back.
The other big problem is that rocket systems have a short flight time, and jet engine systems are too expensive. The jet engine powered backpack worked well, but cost too much. That used a small Williams jet engine. Williams International has tried and tried to make small jet engines cheaper. So have many others. Unfortunately, that's a very hard problem, which is why general aviation is still piston-powered. Below small-bizjet size, jet engines don't seem to get much cheaper as they get smaller. There was a big effort about a decade ago to develop "very light jets", but they ended up costing well over $1 million, most of that being engine cost.
So it can be done, and it has been done, but it just doesn't work very well.