Yes, debt collection agencies can continue to charge interest on debt they have purchased but there are limits to how much and for how long called usery rates. Here is some more information that you may find helpful:
Interest on Debts (when no judgment exists) Section 808(1) prohibits debt collectors from collecting any amount unless the amount is expressly authorized by the agreement creating the debt or is permitted by law. For purposes of this section, "amount" includes not only the debt, but also any incidental charges, such as collection [53 Fed. Reg. 50108] charges, interest, service charges, late fees, and bad check handling charges.
808(2) Legality of charges. A debt collector may attempt to collect a fee or charge in addition to the debt if either:
(a) the charge is expressly provided for in the contract creating the debt and the charge is not prohibited by state law, or
(b) the contract is silent but the charge is otherwise expressly permitted by state law.
Conversely, a debt collector may not collect an additional amount if either: (a) state law expressly prohibits collection of the amount or (b) the contract does not provide for collection of the amount and state law is silent.
808(3). If state law permits collection of reasonable fees, the reasonableness (and consequential legality) of these fees is determined by state law.
808(4). A debt collector may establish an "agreement" without a written contract. For example, he may collect a service charge on a dishonored check based on a posted sign on the merchant's premises allowing such a charge, if he can demonstrate that the consumer knew of the charge.
Interest on Debt (when a judgment exists) States have their own rules for adding interest to judgments. Sometimes called statutory charges, because the amount of interest is set by law. Typically, interest begins to accrue from the date the judgment is rendered until the judgment is paid in full.
Interest on overdue taxes, child support and student loans: The interest on overdue federal taxes and defaulted student loans is set by federal law. Interest on overdue child support is set by state law.
Legal Fees: Legal fees refer to charges for an attorney's services and may also include court costs such as filing fees. Creditors typically add the legal costs of collecting a debt to the account balance. There is a distinct difference between lawyers that are debt collectors and lawyers that are representing a creditor. An attorney or law firm whose efforts to collect consumer debts on behalf of its clients regularly include activities traditionally associated with debt collection, such as sending demand letters (dunning notices) or making collection telephone calls to the consumer, are debt collectors and subject to the FDCPA and must follow the rules outlined above. On the other hand, an attorney whose practice is limited to legal activities (e.g., the filing and prosecution of lawsuits to reduce debts to judgment) are generally exempt from the FDCPA but may still be liable under state collection laws.