Continuing from my comments here #244 (comment), I have an Account 1 that has a balance of 0.999936999999999999 and an Account 2 with a balance of 1 wei that was sent from Account 1 to Account 2 for a transaction fee of 0.000063 Ether.
So then I was going to send Account 2 just enough Ether so that they could make a transaction with gas price of 3 Gwei and deplete their account balance back to exactly 0.
So I sent from Account 1 to Account 2 a value of 0.000063 (which was the transaction fee associated with sending with 3 Gwei gas price), so Account 2's balance became 0.000063000000000001.

But then when I went to create a transaction from Account 2 to Account 1 with a value of 0.000000000000000001 and a gas price of 3 Gwei, it gave me a "You don't have enough ETH balance" error and the "SEND" button is disabled, even though I have enough balance to send 1 wei and to pay the total transaction fee of 0.000063. Note importantly that even though the screenshot below indicates that the account balance is 0.000063, the account balance is "actually" 0.000063000000000001, but that page rounds the account balance to 6 decimal places as pointed out here as potentially being an issue #244