Here is a widely reported Federal Circuit Case, Immunocept v. Fulbright & Jaworsky which discusses, in great detail when a legal malpractice case, brought in state court, and removed to district court may become subject to the law of jurisdictions other than the state in which it was brought.

As an example, a legal malpractice case arising from a simple car accident  [which coincidentally has full diversity of citizenship] may be removed to District Court, but will not be subject to Federal law.  In this contra-similar case, which arose under Patent law, and is subject to federal question jurisdiction, federal law will apply.  Here the statute of limitations was implicated.