Creating Groups, Minimal Overlap

[TomCollins]

Suppose I have 6 letters, A-F.

I want to create 2 groups of 3 for each letter.

Is it possible to create 4 such groupings such that:
Every letter is in a group with every letter at least once but not more than twice?

I'm thinking its not, but can't quite prove it.

I'm creating a simplified version of creating 4 groups of 6 from 24, using 6 pairings such that every letter is matched with every other at least once but not twice. There has to be a good method to this.