Can a sprained hand turn i to a fracture if left untreated after 3 weeks or so ?

No: A sprain is an injured ligament, a break effects the bone. A sprain will not turn into a break without another injury.

No: A sprain is a ligament injury however one can have a negative x-ray that is called a sprain and three weeks later a repeat x-ray may show a fracture that was not apparent in the x-rays 3 weeks prior, a good example is a scaphoid fracture in the wrist may not show up in initial films and is thought to be " a sprain" three weeks later the bone has resorbed from the edges and is more readily seen.

No: A sprain will not lead to a fracture unless you re-injury the area.