"You will not get any expereince that qualifies for a PE" is a rather broad and inaccurate statement.
I and many, many others gained our licenses while working for a DOT.
That said, everyone above is right when they talk about DOT design work moving increasingly to consultants. If design is the be-all and end-all for you, that may not be the job for you.
If, on the other hand, you want to experience more of the real world and not just your structural analysis software, a few years in a DOT can really get you a nice varied background that, as others have said, will make you even more of an asset when you run away screaming to a consulting firm. The trick there is to make sure you get to do enough design that you're still employable by a design firm when you get out.
Another thing to consider is that if you do go to a design-heavy DOT, you might end up encapsulated in their design office and don't get that broad experience after all. But they nevertheless should have cross-training opportunities that you wouldn't get in a private design firm.
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