The perimeter of a triangle is the sum of the lengths of its sides, so if the perimeter of a triangle is 396 inches, then the longest side of the triangle must be less than or equal to 396 inches.
If the longest side of the triangle is 396 inches, then the other two sides must be less than or equal to 197 inches each, since the sum of the lengths of the sides must be 396 inches.
Therefore, the set of numbers that could represent the length of the longest side, in inches, of the triangle is {396, 395, 394, ..., 197, 196}. This set includes all numbers from 396 down to 196, including both of these numbers.