My old 401k could not directly roll over the funds into my new 401k.

Why not? What was their explanation?

Even if they couldn't ACH or wire the funds, they could have still written a check FBO : Your account @ new 401k provider. This would be a check you could not walk into a bank and cash due to the FBO : account listing. Receiving a check of this nature is still a form of direct rollover because you do not have custody of the actual funds. Every 401k provider in the world should be capable of completing this action.

If you requested that they write out a check to you, your name, something you could take to a bank and cash, then that is why they withheld 20%. So yes, you have a 60 day window from the day you had constructive receipt of that money in which to complete an indirect rollover. You must deposit the gross amount of the distributed amount (after deduction of any distribution fee - ex. if you had $10,100 in the account and it cost $100 to distribute, the amount actually distributed is $10,000, from which they would withhold the mandatory 20% for cash distributions) within that 60 day period to another qualified plan. If you do so, then it acts as if it was never paid to you. The tax withholding will simply count as additional withholding. All other factors being equal for this tax year, that would shift you towards the refund side of the equation. The 10% penalty for early withdrawal, if it would be applicable before, would no longer be applicable.

You may run in to issues trying to complete an indirect rollover via a new 401k, especially since that involves 3 parties (you, the plan admin of that 401k, and the recordkeeper). You may have better luck completing it via an IRA as that only involves you and the IRA that you choose.