I asked an acquaintance when she realised her DS1 might be grammar school material (specifically the spectacularly high-achieving coed grammar) and got a bit of an MN answer - 'when he was hopping along the alphabet line in the playground, coming up with a country for each letter'. In Year 1. I gave up on DD at that point. 
However, DD absolutely stormed CATS at the start of Year 5, so we took it a bit more seriously, and yes, got her a tutor. She's not good at listening to DH and me, and really did need the help with exam technique and timing, and a boost with her maths.
She actually didn't pass for the above-mentioned school, but did for the two single-sex grammars we shared results with. Her Year 5 teacher will have seen this coming, post-CATS, but I don't know about the others. And we didn't really - we knew she was was bright, but she's never been a child to come up with some amazing statement that makes you think 'where did that come from?', or to develop an interest and go and learn all about it, or to make great logical leaps ... She's part of a bright cohort, so doesn't stand out particularly. Plus, interesting to read someone above saying that kids who are good at non-verbal reasoning don't really get a chance to show that in school - she is spectacularly good at NVR. Much weaker in maths, so her ability doesn't tend to be apparent.
In short, I wouldn't say it came completely out of the blue, but in my eyes DD wasn't an obvious 11+ candidate. I did have a friend laugh at me and say 'what, you didn't know?' when I was reeling from the CATS results, so maybe I should've just asked her in the first place!