1) In many jurisdictions the ultimate permit and authority come from a state sponsored map of restrictions. For example http://www.dot.ca.gov/hq/traffops/permits/stars.htm (warning: requires crufty Microsoft software).
2) Consider that many bridges cross at an angle, meaning there's more clearance on one side than the other. A height restriction per lane is sometimes seen. Databases like the STARS database above will code more detail than the simple maxheight sign which typically provides a pessimistic view of the clearance.
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