Yes but there are special procedures and time limits that apply to unregistered and unidentified at-fault vehicles.
Claims of this type are brought against the default insurer that goes under the name Nominal Defendant.
The time limit for giving a Notice of Accident Claim to the Nominal Defendant for an unidentified vehicle e.g. in a hit-and-run, is three months. This can be extended up to 9 months if there is a reasonable excuse for the delay but cannot be extended at all after that time.
In the case of unregistered at-fault vehicles, the Notice of Claim must be given at least within 9 months of the accident date but if there is a reasonable excuse for the delay, this can be extended.
In all cases (except for children), proceedings must be started in court within three years of the accident date. The limitation period for children is up until the 21st birthday.
For claims relating to unidentified vehicles, there must be reasonable attempts made to ascertain its identity e.g. a report to the police, taking down names and contact details for eyewitnesses and taking down vehicle registration numbers if possible.
Your unidentified vehicle car accident lawyers will be able to help.