It could be the hammer spring that's not hitting the firing pin hard enough. The 336 has a 2 piece firing pin. If the lever is not all the way up, the locking block will not push the rear firing pin up into alignment with the front pin. This is a built in "safety" that makes sure the locking block is holding the bolt closed before allowing a primer strike,thus firing the rifle. There are one piece light weight firing pins that the cowboy action shooters use, but a one piece pin will let the gun fire before the bolt is closed and the gun in "battery" position.....not a good thing when it happens.....creates the potential for excessive head space as the bolt has not closed fully.......could rupture a case or damage the action, rifle, and or the shooter......the average hunting rifle is better off with the 2 piece factory design IMHO.......
First, make sure the lever is all the way up.
Are you seeing any indentation on the primer where the pin is hitting it? If you are, I'd bet on the hammer spring. If you're not, it's probably the firing pin/pins or the locking block not coming into "battery".
The firing pin is easy to remove from the 336 bolt. Remove the bolt from the gun, remove the extractor, and there are 2 roll pins that will need to be punched out........clean, inspect and replace as necessary......note that the pins are groved to allow the roll pin clearance and the pins have to go back in the same way they came out or you will not be able to get the roll pins driven back in..........not to complicated.........