Frankly I dont remember how many kcal Diet Cola has... and without that I cant tell you, if you can gain weight.
The rule is that is you eat/drink sth too much, much over the level you need as daily calories, it will make you to gain weight.
The question is how much calories Diet Coke has and how much would you have to drink to gain weight.
Beside: it'a all about aspartame, you shouldnt use it, specially in big amount, it's unhealthy.