I do not see the connection between bankruptcy, and the statute of limitations, in your inquiry.
However, if you have unpaid civil damages, not a criminal fine, that are the cause of a drivers license suspension, it may be possible to discharge those damages by filing bankruptcy, and then reapply for your driver's license.
However, there are some types of damages, incurred involving serious injury or death while operating a motor vehicle, that may not be dischargeable in bankruptcy.
General legal advice is offered for educational purposes only. A consultation with a qualified attorney is required to determine specific legal advice as to your situation and applicable law. We are a debt relief agency and we help people file for relief under the bankruptcy laws.
A "Statute of Limitations" (SOL) is created by law and dictates how much time a plaintiff has to initiate a case. After the SOL has expired the case can no longer be filed. The SOL is different for all different types of actions. For a criminal misdemeanor the prosecutor has up to 2 years from the date of the incident to file charges.
There is no SOL on money owed for court fines. You must figure out how to get them paid or dismissed before you are eligible for your license. The debt collection agency that handles 90+% of the courts' collections in Washington is huge and uncaring. What you can do is petition the court to remove your fines from collections and allow you to pay directly. This is pretty simple if you only owe minimal fines to one court and the court is friendly to the idea. It can be almost impossible if you owe a lot of money or many courts.
The law is well-intentioned and is meant to assure people pay their court fines. However, it can also spiral out of control. Someone with one unpaid ticket for several hundred dollars gets a driving while license suspended charge and a bigger fine and the cycle begins until the person is in deep. The law can make perpetual criminals out of otherwise law abiding citizens and doom people to poverty. I have seen cases where people owe $40,000 and have managed to get out of the hole. I've also seen situations where people owe less than $2000 and can't get it paid.
My advice? Don't drive until you have a license and make this your number one priority to pay. Get it paid, get your license and don't ignore any tickets in the future.