Is it possible that a painful breast lump just be normal breast tissue?

See a Doctor. Any new lump in the breast should be a cause for concern, best handled by seeing your gyne or primary care doctor. While most cancers are not painful, it is not a reliable finding. Hopefully, your doctor will be able to alleviate your concern or refer you to a breast center/surgeon for definitive evaluation. I hope its a false alarm--good luck.
It depends. It depends on what you mean by" normal".If you mean that, could it be a benign(noncancerous) process? Yes.If you are young esp. A painful breast mass could be due to several causes which include:abscess, cysts, fibrocystic breast disease, traumatic hematoma(blood clot).All breast lumps should be evaluated by a breast dr or gen surgeon to ensure a benign process.