The washer pulls somewhere near 10 amps maybe more. The dishwasher i'm not sure, but it sould be around 10 amps also. I believe with both appliances on you would pop the breaker.

I know you said it started happening in the last month. It could be as simple as a new device was added to the same circuit. or even the washer getting older and need more current to start the motor. Get electrician be safe.