It is possible to be sodium-deficient if you overexert yourself and lose excessive amounts of sodium in sweat. You can also drive your sodium levels down by drinking too much water and diluting the concentration of sodium in your body.
Both of these things can lead to a potentially dangerous condition known as hyponatremia. With hyponatremia, the extreme loss of sodium can trigger symptoms ranging from muscle cramps, nausea, vomiting, and dizziness to shock, coma, and death.