If you have saline implants, particularly if they were under-filled by your surgeon at the time of surgery, there is always a small risk that a leak could develop and your implant(s) deflate, requiring surgical replacement. Activity is NOT usually the cause; rather, a sudden blow or injury, or something as simple as the repetitive up and down of your chest with breathing, can (occasionally) cause a saline implant to rupture, leak, and deflate. You can't stop breathing, and even vigorous activities don't usually cause any concern at all. Supportive bras are helpful in minimizing movement of the breast implants and breast tissues, and virtually all surgeons recommend additional support when jogging, running, or horseback riding, for example.
If you have silicone gel implants, particularly the present 5th generation of cohesive gel implants in use for the past several years, these implants cannot leak or rupture, regardless of breast movement, and whether or not multiple bras are worn.
However, too tight is still too tight, and if you have painful or tender indentations from your underwire bras, then you should try deleting the underwire bra in favor of snug jog or stretch bras (2 or even 3) when running. Pain or severe indentations in one breast only may mean something is going on on that side, including leak and eventual deflation (saline implants), or if silicone gel, perhaps a muscle strain, sprain, or even partial muscle tear.
My advice would be to tone down the activity a notch or two until the soreness and swelling resolves, and then start slowly (wearing supportive stretch bras as described above) increasing your activity level. If you have saline implants and one has leaked, I would recommend exchanging both for the newest silicone gel implants.