If I was recently run over and received multiple injuries, can I file suit?

It may be possible to settle the case without filing a lawsuit, with the insurance carriers for the at-fault driver (registered owner of that vehicle if the registered owner is someone other than the at-fault driver) and with the insurance carrier for the landlord of the apartment complex where the accident occurred.

You should contact their insurance companies and notify them that you will be filing a personal injury claim.

When you complete your medical treatment and are released by the doctor(s) or are declared to be permanent and stationary by the doctor(s) which means having reached a point in your medical treatment where no further improvement is anticipated, obtain your medical bills, medical reports and documentation of any wage loss.  Your personal injury claim filed with the insurance carriers of the registered owner of the vehicle and landlord should include these items. 

Compensation for the medical bills is straight reimbursement.  The medical reports will document the nature and extent of your injury and will be used to determine compensation for pain and suffering, which is an amount in addition to the medical bills.  Compensation for wage loss is straight reimbursement.

If the case is settled with both insurance carriers (registered owner of the vehicle and landlord), NO lawsuit is filed.

If you are dissatisfied with settlement offers from the insurance carriers, reject the settlement offers and file a lawsuit against the at-fault driver and registered owner of the vehicle, and the landlord.  Your lawsuit would have causes of action (claims) of negligence against the at-fault driver / registered owner of the vehicle and a separate cause of action for premises liability against the landlord.

If the case is settled with one party's insurance carrier, don't name that party as a defendant in your lawsuit.  Only name the party with whom the case has not settled as a defendant in your lawsuit.  If the case has not settled with either insurance carrier, name both at-fault driver / registered owner of vehicle and landlord as defendants in your lawsuit.  If the case has settled with all parties, NO lawsuit is filed.  If the case has NOT settled with any party's insurance carrier, name all parties as defendants in your lawsuit.

If the case is NOT settled, you must file your lawsuit prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.

You can sue someone who was at fault in causing your injuries. That would typically be:

1) The driver who ran you over, if such driver was (presumably) driving carelessly or negligently.

2) The property owner ONLY IF there was something unsafe about the parking lot, such as if you were run over because the property owner did not provide the level/amount/type of lighting that reasonable or responsible property owners typically provide for parking lots.

If the property owner did everything reasonably expected through, such as providing a reasonable level of lighting, you would almost certainly not be able to successfully sue the property owner.