Can an element have many gram atomic weights?

No, not really. An element has a single atomic weight, which is the weighted average of all the isotopes of that element, weighted by their natural abundance. Each element only has a single atomic weight.

What this means is that different atoms of a single element can weigh different amounts because different isotopes have different numbers of neutrons. However, the atomic weight represents the average for all forms of that element, and so there is only one value. The different isotopes of an element certainly have different mass numbers, which is just the sum of the number of neutrons and protons however.