Not sure what you mean? Who is asking them to drop the case? If the client is, then yea, the client can request via the court for a new court appointed attorney. (Or they can cough up some dough and retain a private lawyer). The court can refuse to drop that attorney from your case, but not the lawyer.
If the defendant wants a new court appointed lawyer, they need to have a sound basis for the request of a new lawyer, otherwise the court might not grant your request. Keep in mind that you can't handpick your court appointed lawyer. That's why they call them "court appointed" - because they are appointed by the court to represent you. You can make the request if there is one that you know of, but it is totally up to the court and availability of that lawyer.
If you think the current lawyer can't handle the complexities of your case, then make the request. But if it is just because you had an emotional spat, then you should try to set your feelings aside and make amends. Otherwise you will just be basically waiving your right to a speedy trial because the new lawyer will have to start all over from the beginning to learn your case. Understand that the public defender is on your side. Even though they are paid by the government, they are paid to represent YOU because it is your constitutional right to zealous representation. They will do the best job they can to defend your case - anything less would be considered ineffective assistance of counsel. So make your decision wisely. Good luck.
I agree with gin_in_mi
Generally most law offices do not purchase bad debts.
The collection agencies will continue to own the debts but may turn them over to a law office for collections. It may just be a strong arm tactic on the collection agency's part or it could be a prelude to a law suit.
When a law office is in the practice of debt collection for the collection agency, the law office itself is considered to have a collection agency status. And because of that, the law office is governed by the laws of the FDCPA just as any other collection agency is.
If the law office violates the FDCPA, they can be held liable for the violations just as any collection agency. Plus, they would also have to answer to the Bar Association if a complaint was filed against them for the violations.