The rule is 1 card of each type by each character per check. So on each combat check you may play 1 weapon.
Of course, some cards let you break this rule. A Dagger for instance specifically has a power that you may play it if you played another weapon on your combat check. But, barring something like that, it is 1 card of each type per check.
RotR Rulebook p10 wrote:
During each of these steps, you and the other players may perform only the specified actions. Players may only play cards or activate powers that relate to each step. Each player may play no more than 1 card of each type during each step; for example, no one player may play more than 1 blessing while attempting a check, though multiple players could each play 1 blessing during that check. Each player may activate any power no more than once during each step. Players may not play any cards or activate any powers between these steps.
RotR Rulebook p11 wrote:
Attempting a check requires several actions that are explained below. Remember that each player may not play more than 1 card of each type or activate any 1 power more than once during each check.
The reason some weapons are 2-Handed is other cards you could play will care about that. Many of the armors (Shields) for instance say you can not play them if you played a 2-Handed weapon on the check.