Hello, welcome to JustAnswer! I am a Veterinarian and will help you right away!
They definitely can.
They especially like to tear or break the dewclaw which is like the thumb for a dog, or the big toe if on the rear limbs.
What happens usually is the nail gets caught on something, or the pet puts the brakes on a little too fast, and the nail breaks.
Typically, the best thing to do is numb the nail, and remove the broken section. Otherwise it is just plain painful constantly. It will grow back over time as long as the base of the nail (ungual crest) where it grows is not damaged.
It is also possible to fracture a toe (complete or hairline), to have fungal and bacterial infections of the toe, and even for there to be tumors
present in the toe.
So, it is a good idea to get it checked out to be sure.
Broken nails or jammed toes will sometimes get infected. You might be able to prevent that by soaking the affected foot in dilute betadine and water (to a weak tea color) for about 5-10 minutes twice daily.
For pain you might try aspirin at 5mg/pound every 12 hours. Buffered, not enteric coated.
If there is swelling in the toe already, definitely have it checked. Some toe cancers are extremely agressive but can be stopped if caught early.