I get 1 in 20.
For all three to have the same sum they must be the pairs 1,6; 2,5; 3,4. As the pairs can be in either order you can have any card in the first position of each pair. If you take the 1st, 3rd and 5th cards as the first card of the pair to represent the pair, you end up with a combination of choosing 3 cards out of 6 which gives 1/20.
As I've grown older, I've found that my memory is not as good as I used to think it was.