I read somewhere that If you have a Savings Bank Account and the interest earned through Savings Account throughout the year is less than $10, then Bank will not send you the 1099-Int, so you don't have to add this as your income on 1040, is this correct?

Yes, and no. The bank is not requried to send out 1099-INT forms to people who made less than $10 in interest. But you are still supposed to report the income. Of course, without benefit of a 1099-INT there's no way the IRS would know if you don't report it...

You can be morally correct and report the income, and still pay NO additional tax, because tax assessments normally increase with each $50 increment of income. If you add about $7 to your income level, the odds are you will see no additional tax due.