Assuming we are in post-apocalyptic wilderness where herbs are readily available for treatment, would it be feasible for someone to be stabbed with a knife and not only survive but heal enough to keep moving within a week max? The wound can be anywhere, but I was thinking the only place it could realistically be is the upper arm around the bicep.
A stab with a clean blade may not be fatal and heal almost perfectly, even without modern medicine and outer help, if the blade travels only through skin, fat and muscle, possibly cutting some vein, but no major arteries.
If anything else is in its trajectory things get stingy, unless you have a doctor available.
A modern E.R. is not necessary (but it helps); doctors in Magna Grecia were perfectly able to help non-desperate cases and so were western doctors even before the invention of antibiotics.
A stab in the guts would have been almost certainly fatal.
In post-apocalyptic settings you can assume the presence of "doctors" akin to what they had in the Far West, I presume.
A relatively short and skinny blade would be the best tool to get stabbed with.
You'll want to avoid organs, so the abdomen is out. The arm would be a good place like someone suggested. Possibly the character was trying to protect another part of his body and blocked the knife with his arm or fore arm. As long as you avoid the under arm area, because that's where major arteries run and if those are punctured it would most likely mean death.
The thigh is another good place. It's thick with muscle and being pierced with a smaller blade would probably be just a surface wound. The back of the leg though has major arteries and nerves that, if damaged, would require more than home treatments to fix. I also suggest looking at human anatomy diagrams to find ideal places to put this stab wound.
A stab to the face (save the eyes) or scalp won't hinder your character's ability to run or use the arms. Sure the scalp bleeds like stink, but you know, that's what cautery irons are for. If blood loss isn't too great, could be running around within an hour or two.
Regardless of where the stab is, the primary factors in play will be luck and even more luck. A (relatively) clean wound that is quickly irrigated with (relatively) clean water and treated with (relatively) clean instruments and bandages will heal well enough on its own.
Getting up and moving again --- that will largely depend on how dire the need is to get up and move again. If your character is on the run or otherwise needs to get from one place to another, well, she'll get up and move.
So "realistically", your character can be stabbed just about anywhere that does not involve a great vessel or vital organ or needed tendon (Achilles tendon, anyone?) and heal without a well equipped modern ER / OR. People have been stabbing each other, biting each other, being attacked by predators and so forth for millennia and for some reason we still seem to be around as a species. Healing may not be ideal without proper care, but it does happen.