I've paid three different people within 1 minute using 3 separate transactions on my Electrum client on many occasions, and never had a problem.
The only way I could imagine it ever having a problem would be if you tried to send 3 different transactions within milliseconds of each other, and the first transaction didn't have enough time to go to the bitcoin network and come back (it needs to do this to check whether the network will accept your transaction or not) and so the 3 transactions you made are all referencing the same bitcoin (in other words, you inadvertently made 2 double spend attempts, and the network rejected them)
Other than that, the only thing I could say would be to post screenshots of the transaction details screen of each transaction... then we can see what's going wrong maybe?