Well, the obvious assumption or problem in concern into an evolution problem is:
1) Evolving Pokemon is holding an everstone or
2) You pressed the B-button
But, I've played Pokemon Plant and Pokemon Revolution Online - similar online games that use Pokemon Red, Blue and Green as the storyline's background. In this case, you may want to consider that the person you are trading your Kadabra to doesn't have a full team of 6 Pokemon.
When you look at newer generation games, when you trade, you can trade any Pokemon from your PC, regardless of it being in your party. However, in older generation games, if the opposing person receiving the Kadabra has a full team, Kadabra is directed to their PC, preventing this evolution.
So, you may want to try this. Hope it works! :)