Reward card eligibility, previous open cards

[Allen]

So,

I understand that to get some of the neat offers like $400 cash back after spending "X" require you to be a 'new' customer.  I understand that 'new' means you haven't been a customer of that bank for "Y" amount of time.

How long is "Y" generally?

Also, in going through my credit report, I see I have some old open credit cards with companies that offer these cards.  I assume I need to call and cancel them, wait "Y" then I can apply and get the rewards?  I understand there will be some minor hit to my credit for closing these cards.  I'm OK with this and haven't used them for years.  However, one has an available balance of $21.6k.

Any advice or concerns?  My credit is excellent (estimated 780 FICO by CreditKarma)

[chicagomeg]

Most of the time you don't have be a new customer of that company, just a first time customer of the card itself.