saru mo ki kara ochiru
Be careful with how and when you say this one, because it can sound like a warning, or a threat, if you say it to someone who has not yet screwed up.
If you say it AFTER someone makes a mistake, however, and say it kindly, it's interpreted as a comfort.
Literal- Even monkeys fall from trees
1. Everybody messes up sometimes.
2. Pride goes before a fall
The nuance here is that it should be used when someone has messed up at something that normally, they are very good at, OR is natural to them. If you ever happen to accurately correct a native speaker's Japanese grammar or kanji or something like that, it would be a great time to use this, feign modesty, and impress them even further.
A bonus phrase that goes along with this expression, particulary the "pride before a fall bit," is the following:
Your 調子, as most of you already know, is your condition, or state of health. You can use 調子 to ask about someone's health (調子はどうですか？) or the condition of other things, like machines (車は調子悪い。), but did you know you can ride on it? "Riding on your own condition" is a Japanese way of calling someone stuck up, arrogant, or all puffed up on how cool they think they are.
We've taken to reminding each other and our Japanese friends during wakeboarding sessions (when someone invariably tries to do a cool trick and ends up eating it really hard): 波を乗り、調子乗ってんじゃね！Ride the waves, not your choushi.
Nirav didn't know the meaning of 「切磋琢磨」? Well, everybody slips up sooner or later. And anyhow, messing up every once in a while is good for that giant ego of his, don't you think?