Does internet connection affect online gaming?

[Q]

I play a lot of Halo 3 and CoD, and a lot of bullets seem to vanish into thin air. I have Verizon DSL.

In Halo 3, it takes 12 bullets from a BR to kill a player in standard matchmaking. Many times I find myself hitting them with far more than 12 shots. Is this due to MY internet? Or is online gaming in general just slow? Today I watched a Halo 3 game in theater mode and watched very closely. I saw 15 bullets hit the enemy; 5 bursts, and every shot hit... The last 3 which were in the head.

In CoD, I shot 4 shotgun shots at an enemy, yet he killed me. When I watch the killcam... It says nothing fired from my gun? This even happens when I have host... so I don't know wtf is up with online games.

So does this happen to everyone? Even people with elite internet connection?
It's not lag. The games run smooth, but bullets just don't register sometimes.
The fact that you guys don't know what I'm talking about eaither means:
1. You don't play these games
2. My inernet sucks.