1 person asking for $1000 in cash isn't going to make a dent in a bank. 1000 people withdrawing $1000 might be a problem. I have personally taken $5k out of my bank in cash upon asking for it without a problem and I'm positive I could do it anytime. You vastly underestimate the holdings of a bank, they are required to keep a a certain amount of cash on hand just in case. That's why big banks have big safes, and small banks don't.
I know of two banks in my country, the largest two, who will most likely tell you you have to notify them atleast one day before you wish to have your money in cash. They will also ask you what you intend to do with such large sums of cash (might avoid this question of youre a native and wear a suit and tie).
Although I have been in a bank where it was no problem taking out $2000. I guess this depends on banks.
Not long ago, while at the queue, I could take a glance to a paper describing the monthly movements in that branch's vault. At no moment the balance was lower than 70k. Since I know nothing about the banking world, I'm leaving open the question about it being "enough" (for what?) or not.
I usually withdraw 1200-1900 (at least once a month) and I've never had any problem with that.
Oh, this is Spain BTW.