Can the sum of the magnitudes of two vectors ever be equal to the magnitude of the sum of the same two vectors? If no, why not? If yes, when?

They cannot be equal

Explanation:

Let 2 vector equal in magnitude and opposite direction. One of them is 1 and the other is - 1

The sum of their magnitudes is 1 + 1 = 2

The magnitude of their sum of 2 vector is 0 since the 2 are in opposite directions, they cancel out each other and their final magnitude is thus 0.

So the sum of the magnitudes of two vectors cannot be equal to the magnitude of the sum of the same two vectors.