I apologize if this is the wrong forum to be posting this in, but I figured since it's not a 'game breaking' issue it would be best suited under general discussion as opposed to hardware/software issues. Anyways, here's the problem i've been having:
Some of us, myself included like to play a kind of sniper type of character, always have when it comes to the Fallout series. Anyways, I play through the intro, I figure stuff out, then I head on over to Primm. Talk to the NCR, find out what's going on, and i'm on my way into sneaking into town sniping escaped convicts left and right! UNTIL, I sneak around the sheriff's shack to see two convicts out in the open(in between the casino and the hotel) so I load up my rifle and sneak attack one of em', one shot one kill. So i'm taking aim at the other guy and all of a sudden my status goes from hidden to caution, so I creep back behind the building for cover and wait until it says hidden again, when this happens I sneak back around and start taking aim, all of a sudden back to caution...so at first I think i'm too close...so I go a little bit further out, and the entire time, the status is going from hidden to caution hidden to caution all in about five second intervals. Then I realized what was happening. Escaped Convict #2 noticed his dead friend, ran over to him(hence the caution) after a moment or two of not seeing anything starts to walk away...then notices the body again, back to caution, and so on and so forth, just back and forth for like five minutes before I decided to just run in guns blazing!
So i'm just wondering...has anyone else noticed this issue? Like I said, it's not game breaking, but it's really annoying especially if your specializing your character in stealth.

I don't personally consider it an issue. To be honest, if I saw my buddy capped, and I didn't know why, I'd stop "patrolling" all together and run for cover. Barring that bit of AI, I'd expect them to be a bit more cautious, EVERY time they saw their dead buddy. Woudn't you? You see your friend's brains all over the pavement, you wouldn't get nervous? I would! But again, I'd hide. Snipers are deadly.

Yeah, that happens occasionally. I deal with them either with explosives, or VATS a headshot at the moment status returns to Hidden.

Try this same tactic, but move slower and see if he alerts again. I'm on PC and one of the only gripes I have is we have two speeds: Slow and Too Damn Slow. But when we have the cap lock on, ie: we are running, the NPCs will alert real fast. Toggling the caps lock so I'm at a walk (slow death crawl) speed usually makes a difference. I don't know what the game does under the hood, so to speak, but perception and recent alert status may have something to do with this. Deathclaws and Super Mutants seem to take forever to get back to where any movement at all doesn't instantly put me right back at [Caution].

-Gunny out.

Fellow sniper here. I've noticed the same thing, almost as if their AI program is caught in a loop or something. Doesn't happen to me a lot, but it's happened enough that I remembered it. I would say "what if you time your shot so you pull the trigger when the 'hidden' is showing", but you can't see the hidden/caution thing while looking down a scope. I just nail 'em once, then waste a few more rounds finishing them off. And if I'm far enough away, the signal never gets to 'danger'. But at the range you're talking about, I'd probably pull out my back-up weapon, and get ready to "Rambo".

Edit: Scratch that, that's still a ways away, they'd have to be closer for me to pull out my back-up.

Walk don't run, and if low in skill under 40 maybe even 60 don't move at all after the first kill.
The first reaction of npc's is to head towards the kill you made or last shot, then they'll scout out and head in your general direction.
If well hidden and or a high sneak the caution will go away completly.

So the general rule of any sneak in Bethesda / Obsidian use of Beth engine game..
Is scout, sneak, cover, target, shoot, stay still, wait, line up second target wait till "hidden" shoot again.

Silenced weapons also make this a lot easier, a noisy kill will keep them more alerted.