Can a blood clot cause painful swelling to occur in both legs/ankles?

It's possible: Usually clots are more likely to appear in one leg or the other, appearing in both legs simultaneously is less common. However, it is possible to have clots in 2 legs at once. You should contact your primary care doctor or go to your local emergency room for an ultrasound to rule out a deep vein thrombosis or "blood clot" which can be life threatening.

Both legs swelling: It would be unusual for a clot in one leg to cause both legs to swell unless the clot is located in the inferior vena cava which drains the blood from BOTH legs. Often times bilateral leg swelling is due to other issues such as heart, kidney, lung or liver. Even medications can contribute to bilateral leg swelling. See your PMD for an evaluation.