Yes, generally you do have 3 years (called the limitation period) from the date of the accident or injury in which to bring a claim (meaning that you must either settle your claim or issue proceedings in the Courts within 3 years) but there are exceptions to the 3 year rule.
Any Personal Injury Lawyer that does not know the rules relating to limitation is heading for big trouble so make sure you speak to a good and experienced Personal Injury Lawyer. Keep in mind, the main source of negligence actions against Personal Injury Lawyers relates to the Limitation Period.
This article provides you with only a brief overview of the law relating to Limitation and is by no means exhaustive. So, as I have said above, for the very best advice relating to your claim speak to a good Personal Injury Lawyer.