Quote from CVS Misfit Doll Toy - "I'm a little rag doll who just wants a friend. I think that will help my broken heart mend".
The producer of Rudolph (Arthur Rankin Jr) says Dolly's problem is psychological depression after being abandoned (same answer as in Wikipedia). Before this, she must have been rejected by every little girl.
The one who abandoned her might have gotten her as a Christmas gift but didn't want her either, neglected her, and finally tossed her out. Dolly's depression was likely SITUATIONAL, caused by no one wanting her, feeling hopelessly unloved, and being abandoned (a.k.a. Post Traumatic Stress perhaps).
Going back to the CVS Misfit Doll quote above, the only thing "broken" about her was her broken heart caused by no little girl wanting to befriend her.
But Rudolph and Santa find her a little girl to belong to, and Dolly winds up being the first toy delivered that Christmas Eve (shown just before the credits roll at the end of Rudolph).