You need to be a much better positional, pre & post-flop player to succeed in 6-max 5-10 relative to full ring (even at some higher stakes) games. e.g. $10-20 full is an 'easier' game in that sense.
I emphasize post flop play because you have to know when to make plays like check-call someone down with middle and even bottom pair at times, or frequently make semi-bluff turn raises on someone when you think they have top pair bad kicker 'cuz they are capable of playing that hand and laying it down. You also have to be able to make per player adjustments better. Handling aggros and maniacs and rocks alike all play a part in 6-max success.
A word of advise, don't even think about adding multi-tabling in 6 max until you fully adjust and can prove to yourself that you can beat the game. And even then, add tables slowly, but I wouldn't recommend more than 2 or 3 personally. If you play tigther you can play one or two more, but if you are a positionally-aware, opponent-aware chameleon lag/agg type, then I'd never play more than 3 at once and even that can be too much. Reads and playing the player are just too big a part of the game IMHO . Multi 6max before you are ready, and you'll drop way more than then that $1000 Fnord and KMD are warning you about.
The reason you need more that 300BB is partially attributable to you having to make adjustments to your game (some people adjust quickly, some never do) and of course mostly related to the natural increased variance of that game structure.