Answer:
Depending upon whether you are doing this for personal or business reasons, the sky is really the limit. Typically, most people only have 1 savings account per bank, but businesses have multiple accounts for each product line.
The FDIC limit is $250,000 per depositer per bank, so you would want a little less than that in any one account, otherwise you increase your risk exposure. You can generally negotiate a better interest rate with a higher balance, so the more money you have in one account up towards that limit, the better.
Savings accounts are terrible investment vehicles, but until you have socked away sufficient capitol to find a better return, they beat mattresses.