Tough question.
One way of thinking would be that you need as much broad experience as you can get before you take the PE exam....not just for taking the exam, but to properly prepare you to be a licensed professional engineer. Being a bridge inspector would not give you the broad, nor in-depth knowledge and experience, that a PE needs to have.
On the other hand, I've seen folks who have gone to work for municipalities, or the DOT, or pre-engineered building manufacturers, and only gotten a very narrow experience history over 4 or 5 years, sat for the PE exam, passed it, and are now licensed professional engineers.
So I guess I'm saying it works both ways.
With either case of experience, you will need to know when you don't know enough to be designing or taking responsibility for something, even though you will be a PE. You will need to know your area of expertise and knowledge and be careful when determining if an engineering task is appropriate for you or not. For example, I'm a consulting structural engineer, but, if someone brought me a high rise building to design, that's something that I have not worked with and would send the project to someone with high rise building design experience.
Hope this helps.