Can you make a partial payment to AT & T without getting your service interrupted?

Call and ask them. If you are upfront it is less likely to happen.

It probably depends on how far behind you are, how much you owe, and how much you’re willing to pay. If it’s a mobile and you have a lot of extras, you might also negotiate a reduced plan for a certain time, just to guarantee you have basic phone service.
If anyone on the service depends on the phone for their health or livelihood, that’s also a consideration.

As @ARE_you_kidding_me says, only AT&T can tell you for sure.

@ibstubro is correct about the next steps.

But most companies will accept partial payment as evidence of your intentions, and won’t cut you off. They may assess some fee or another.

But paying something is much better for you than paying nothing.

A guess on my part, a partial will keep your service turned on. But i think they would want a payment schedule to get current, miss a payment and no more phone. Here’s their late payment page

Oh I think AT&T will charge for both cancellation and re-activation. Keep the phone on.

Negotiate this with a supervisor rather than a first-line customer service rep.

Someone at a higher level has much more authority to make payment plan arrangements rather than the first person who answers the phone who typically can only tell you what CAN’T be done.

Get bumped up to a supervisory level in the bill payment division. You may be on hold for a few extra minutes, but it will usually be well worth waiting for the right person to speak to.

You’ll have to call them. Unless someone here is a service rep there right now, no one here knows. They change that policy from time to time. The thing that doesn’t change is the need for you to contact them if you are going to be late paying your entire amount due.

AT&T is renowned for shifting calls to CS agents around the country. Although ‘possible’ to see notes from other operators from previous calls, you will not get definite answers about any action taken from those previous call inquiries. Always get their name, location, and their manager’s name. Friends have had operators disconnect calls or have been put on an hold with no intention of getting back to the call. One friend had cities, times, dates, names, stupidvisor’s names, etc. – it took seven calls to get someone to try to correct the errors, eleven until it was completed. Have care if ‘transferred’; to whom, an outside phone number if disconnected, etc. He switched carriers afterward.
Sad statement about a company which used to have a pretty good CS history.