Yep, you can OD on nicotine, but nope, you won't do it from vaping the standard mg. You'll get sick waaaaaaaay before it gets lethal. I was chain vaping (@18mg) when starting ecigs, got myself woozy quite often. I'm talking chain vaping non stop for 1-2 hours straight. Stopped, took break sometimes lay down till they passed, and started up again. I have some other issues that lead me to be a bit excessive with things at times though, and was pushing past the initial warning signs.
You start to know the signs, the physical sensations from it, they're very distinct and you'll know what they're from. There's no mistaking "Oh crap I think I have had enough for now." Everyone has their own tolerance, you find your own and naturally adjust. It's not a fun sensation at all, not something that you go out and do when out on the town to have fun.
Nicotine IS highly toxic, but movies are movies and you'd have to intentionally try to kill yourself (or be doing unsafe DIY practices) to go that far.