When it rains, it flows—a nasty soup of stormwater and sewage spilling into local streams.

Controlling these sewer system overflows costs a lot of money and can require tricky engineering: some cases require the construction of large storage tunnels burrowing hundreds of feet underground. Many cities want more time to clean up the problem, and they’re getting it.

Washington, D.C., is among the cities that have taken advantage of the EPA’s integrated planning policy, which outlines how communities...