I recognise this will vary. When we were broke uni students, a grand or two in the bank qualified as a little nest egg. These days, it's probably something in the five-figure range.
The other day, though, I was carrying out a mystery shopping assignment at a bank. Part of my fake scenario was having 'no real savings', which was specified as no more than $30k max.
Call me crazy, but is that not sufficient to qualify as 'real' savings? That is a significant chunk of money. Particularly as I was also supposed to have a ridiculously small amount left on my fictional mortgage ($200k) so it's not like I'd be saving hard for a down payment in this scenario. And, if you had much more than $30k, you'd most likely be investing it rather than have it sitting in a savings account anyway.
I tell you, I felt ridiculous sitting across from that bank staffer. I wound up specifying that I had $10k in the bank (the best lies are those that stick closely to the truth!) which she sort of chuckled/snorted at and said 'that's a lot!' And I felt like apologetically adding 'yep, had some help from the parents' or 'married a rich dude' since there's no way I would have that small of a mortgage balance at this age. It didn't help that I think she thought I was even younger than I actually am.
If I'd been in her shoes, I would've hated me so much. She scored awesomely on the assignment, so at least there's that. And who knows, she might have found out that I was the surveyor, and not a real customer.