To recover bodily injury compensation a plaintiff must prove not only that a defendant was negligent, but also that the defendant's negligence caused the plaintiff's injuries. Thus, the defendant in your first car accident case might argue that you are unable to prove which of your injuries were sustained in your first car accident, as opposed to your second car accident, and that you therefore are unable to prove your case.
Assuming your "Accident 2" injuries were sustained to parts of your body different than those injured in "Accident 1," proving which injuries were the result of Accident 1 shouldn't be an issue. In fact, it is often possible to prove apportionment even where the same body part was injured in both incidents. This is frequently accomplished by comparing medical imaging scans (such as MRIs) that occurred between the first and second incident with scans conducted subsequent to the second incident.
A more interesting issue arises if you are able to prove you injured the same part of your body in both Accidents 1 and 2, but unable to prove the extent to which the injury is attributable to each incident. In this situation the burden of proof as to apportionment shifts to the defendants. If the defendants are unable to prove the extent to which their respective negligence contributed to your injuries, then each defendant is jointly and severally liable for the entirety of your injuries (assuming liability can be established as to each defendant).