Can you sprain the bottom part of your foot by your toes

Yes: Since a sprain is defined by a tear of the ligaments that connect the bones together, any ligaments can be sprained. The ligaments around the toes are easy to sprain, but they are also easy to heal in general. If there is profound swelling or no improvement with inability to put weight on the foot, then i would have it checked.

Sprain possible: There is a lot going on in that foot of yours. It takes a number of muscles and tendons to move the toes and hold them stable which walking. Any of the muscles, tendons or ligaments can be bruised or sprained. The specific diagnosis has to do with where it hurts and with which activities. In the meantime rest, ice and well-made shoes can help.