Can one node receive two transactions from the same user (doublespend)? I'm learning how the blockchain works. I've read that when someone (say Alice) send twice a transaction to two different peoples (Bob and Charlie) such that Alice has only enough to pay Bob or Charlie, not both, then the first validated block containing of the two transactions will be accepted and the other will be rejected.

My question is : What happens if the two transactions Alice -> Bob AND Alice -> Charlie are sent to the same node and that node validates its block ? Is it possible ? If not, why ?
No that would not be valid, the node would reject the second transaction it saw, because it would know the UTXO was already spent by the first one. If a miner tries to include both in a block, that block would be invalid too for the same reason. 

The exception is if the first transaction signalled to opt in to Replace-by-fee and the second transaction had a high enough fee to replace the first one. (BIP 125)