This particular question will have 2 different answers depending of the time period & the rule books that you are using throughout the evolution of the game.
In the Parker Brothers/Hasbro rule books of 1959, 1963, 1975, 1980, 1993, 1999 & 2016 you will find the following:
If during your turn you eliminate an opponent by defeating their last soldier on the map, the eliminated player is out of the game & you win any territory cards that player has collected. If winning them gives you 6 or more cards, you must immediately trade in enough sets to reduce your hand to 4 or fewer cards. This is called a mid-turn trade-in.
In the Parker Brothers/Hasbro rule books of 2003 you will find the following:
If during your turn you eliminate an opponent by defeating their last soldier on the map, the eliminated player is out of the game & you win any territory cards that player has collected. If winning them gives you 5 or more cards, you must immediately trade in enough sets to reduce your hand to 4 or fewer cards. This is called a mid-turn trade-in.
Since 2003 & prior to 2008 it is impossible to have 6 cards in your card hand at the beginning of your next turn as you must trade them in until you have 4 or less cards. Your earning another card at the end of your turn can only bring your card hand to a maximum of 5 cards at the beginning of your next turn. There is no confusion since 2003. In 2008 Parker Brothers/Hasbro moved onto a Star method of trading in cards rather then the 3 image method. In the Star method of trading in cards you can trade at any time during your turn always & you can hold onto more then 5 cards. In 2016 Hasbro returned the mid-turn card trade-in to 6 or more cards so the confusion has returned.
Pre 2003 & as of 2016, it is possible to eliminate an opponent, get their cards bringing your card hand to five & then ending your turn earning your sixth card. At the beginning of your next turn you were forced to trade in at least one set. If you wished & were able to do so, you could trade in 2 sets of cards at the beginning of your turn.
The only way to end up with 6 cards under any set of rules while using the 3 image method is by eliminating an opponent & earning their cards. Therefore your friend wishing to wait until he had 6 cards to trade in a set could not do so under any set of rules because if you have 5 or 6 cards at the beginning of your turn you must trade in at least one set. If he had 5 cards at the beginning of his turn he had to trade them in.