Heavy downpours in South Texas washed out highways and stranded motorists Tuesday as forecasters warned that a threat of more severe weather could bring dangerous flooding to already drenched counties near the border with Mexico.

Slow-moving storms could drop a summer’s worth of rain over parts of Texas this week, bringing a significant flash flood threat to the state.

Parts of southern Texas are under back-to-back days of a rare “high risk” of flooding rain, spurring dozens of water rescues, as more rounds of slow-moving storms set off…