My own story:
When I first graduated many years ago - I first worked for a Civil/Structural Consulting firm that primarily focused on highway/civil projects with some exposure to buildings -I was a field inspector for numerous bridges.
After graduate school I worked for 9+ years in a Structural firm doing nothing but buildings (only an occasional "bridge" but nothing with DOT's).
Then switched jobs and began running a structural department that was responsible for both buildings and bridges. I hired some young engineers and trained them in both (I basically had to learn bridges myself).
This was a challenge as the bridge designs followed USA state DOT guidelines, standards, and AASHTO which all had to be kept up with. In addition, the UBC, BOCA, and SBC all had to be known and then came the IBC.
After about 6 years or so of doing both, we split our department up to better market and focus on the two specialties. We were successful in doing both, but it was a struggle to be able to market DOT's and stay up with their business and also to keep up with the now fast-changing world of building codes and specs.
It can be done. But not many firms do both.