It's more likely that you have a problem with the washing machine. It's evidently drawing a lot of power, thus reducing the voltage and dimming the lights. I'd have it checked out, because a washing machine shouldn't do that. It's highly unlikely that this is a breaker issue.

Is it a momentary flicker when the motor starts, or a discernible dimming the whole time the washer is running?
The former is probably normal, the latter definitely is not.

Standard loose/corroded neutral issue.

Central A/C doesn't do it because the compressor is 240, and doesn't use the neutral. Washer does.

Typical advice is for OP to call qualified electrician to inspect neutral connections at panel and meter base.

If the problem is limited to the washer, then I respectfully disagree with the diagnosis that it's a neutral issue. I would call a washing machine repair person. To me, it appears that the washing machine is drawing much more current than it should for brief periods, causing the flicker (but not tripping the breaker). Probably worn out contacts in the timer.

I made a suggestion based on the probability of failure. It's highly unlikely that an intermittent neutral will manifest itself only through a washing machine. That being said, I'm not at the OP's location, so I can't give a precise diagnosis.

Did you miss your prunes today?

I've experienced the same issue with the lights dimming during the agitation cycle. I assumed it was caused by the 20 years old Kenmore washer that came with the house. I have since replaced the washer with a brand new Maytag, and lo and behold the same issues occurs when this washer enters the agitation cycle. I don't believe it to be a hardware issue with the washer, since two have done the same thing. The only other thing, is the outlet and or breaker.

Based on a previous post, I'll be checking the neutral lead, to ensure it is making good contact, the outlet is wired correctly and the ground it good. I'll post my outcome in a few weeks.

I have had the same problem your describing. I live in southeast louisiana . Flickering lights thruout house when the new washer was on agitation cycle. Has done this for years. My dad and i could not find any cause in the house: breaker connections no, sockets no. Called power company . They sent a crew same day and found an open neutral buried under ground. Dug it up and fixed it that very afternoon. No more blinking case closed.

I have the exact same issue but performed a few different troubleshooting steps that were not mentioned here. To begin, this is a rental home built in 1972. All aluminum wiring with some copper for lighting overhead that was updated in 2009. My washer is brand new, LG high efficiency & overpriced. I’ll add that I have a matching oversized LG gas dryer too. They both plug into 1 receptacle with the a top and bottom outlet. This is wired back to the breaker box as 2 20 amp circuits. They share a neutral wire. The circuit breakers were old and not joined with any bar that physically connected the two switches into one so that one off makes both off. The receptacle was old, loose and burnt up some. And even more concerning, the tab on the line (right side) was not broken to properly separate the top and bottom circuits. Luckily I didn’t find out the hard way. Although the panel was not labeled/correctly, I went through process of elimination by turning everything off and then testing one breaker one at a time and matched to outlet(s)& lighting it controlled. I assume that someone replaced this laundry receptacle at least once in the distant past and forgot to break the right tab. I replaced the receptacle with a new 20 amp outlet and broke the tab on the right side only. There is a red & black wire (on right with tab connecting top to bottom removed) and one shared neutral wire that is white (on left with tab intact.). It is also grounded of course. I also replaced the two circuit breakers in the box with new and specifically for this type of setup. It has the bar that physically joins the switches for off and on. I also checked and tightened all neutrals in the panel. But even after all that, the damn dimming still occurs when agitating. I should add that all my lighting is LED too.

Here is the last important clue that I would think others could try too. I was doing some laundry and it had just started to agitate and dim like always. Just as a temporary measure, I paused the cycle & then hooked my washer up with an extension cord to another nearby circuit. It’s about ten feet away and it’s a heavy duty cord but was my last straw of an idea to check & was not a permanent arrangement. Anyways, the washing machine worked fine and there was no dimming throughout the entire cycle. I left it like that for a few days and it continued to perform without dimming the lights. Additionally, this dimming would occur whether the dryer was also being used or when it was off. Keep in mind that they share one neutral. I tested both the top and bottom outlets and the readings all indicate a grounded correct setup. There is no open neutral.

I still don’t know why it does this but believe it is definitely related to wiring, not machine. Perhaps i also need power company to come out like person above had when their issue was resolved. I’m moving to purchased home in a matter of weeks so the landlord can decide how to proceed from here.

just to give you an idea of the mess this was....

We were having the dimming issue most noticeable when the washer was agitating. Even plugged into a different circuit and found the same thing. Called out our power company, turns out the neutral was faulty coming from the transformer to the house. ( we have buried service). Fixed in 10 minutes, no hint of dimming now. Call your electric company, it's free and potential safety problem solved