Yes, both. Still flickers.

What kind of bulbs are these?

Led

So...have you checked to see if the LED bulb is inserted fully? And have you tried inserting a different bulb to see if the issue is with the bulb or the socket?

Is there a dimmer involved? I've seen LED bulbs "interact" with each other even if the dimmer switch is on the same circuit (but controlling a different load).

No dimmer. Yes, i have taken this bulb in and out and switched it with another led bulb. None of the bulbs in the other sockets of this fixture flicker.

I'm thinking one of two things ... either the bulb is not making good contact with the hot tab at the bottom of the socket, or the factory splices joining the four sockets together are no good (one or both). I've come across the latter several times.

Yankee, if the latter, then should be returned? Is there any safety issue with it as is?

You say you remodeled the bath. Did you select and install the fixture? if an electrician did, then have him replace it. If you did, then remove it and return it to the dealer for exchange. Make sure you do that with in the dealer's "free unquestioned exchange" policy, otherwise you have to prove to them that it is defective.

What mtvhike said ^^^.

As far as it being a safety issue, I doubt it but I can't say for certain without seeing it.

I selected the fixture, and had it sent to the contractor who installed. It came from Fergusons.

Update- i hAve asked two other people to look at the light to see if they think it is a problem and neither one can see this! Both are over 60. Is there something wring with their vision or mine? Ever heard of something like this being visible to some and not others? It is not a constant flicker, just when the light is switched on.

LED lights do that. They can flicker or be a little slow to start. I have two hall lights on the same switch, one LED, one incandescent. You can definitely see that the LED takes longer to turn on than the incandescent.

I have different brands of LEDs in a bathroom multilight fixture for experimentation. They all start at different times. I have some go bad prematurely and start-up is one of the indicators (eventually they don't).

If you still see flicker there is a lose connection possible with the factory wiring to home cabling and switching

There are a couple reasons this happens. The house electricity is 60Hz alternating current. With a standard incandescent bulb, the filament heats up to create a glow to produce light, which there is not much visible effect by the power turning on and off 60 times per second. On the other hand, an LED does, in effect, turn on and off 60 times per second, but you usually can not notice it since it is very fast. Just like you do not notice the individual 60 frames per second on your TV to produce moving pictures. Inside the LED bulb, are Solid-state components (passive switches, unlike the wall switch that you have to toggle to turn on and off). The electricity will only go through an LED in one direction, therefore, there are rectifying diodes inside the bulb to make the AC current flow in the correct direction, and capacitors to regulate the flow of that electricity. These Solid-state components need to be "charged" before they can work properly. In mass produced bulbs, especially the cheaper ones, these components can be slightly defective, but still work, or the "charge" may drain out between uses.(you may have heard that you should not work on the electronics in a TV, even if it is unplugged, because it will still hold a charge) A drained charge, or slightly defective component can cause a flicker at startup. If you have switched the bulbs around in the fixture, and it is always the bulb in the same socket, then one or two inferior bulbs should not be the main cause, but can still contribute some as a quality/engineering issue across the board of the brand/model# of bulb.

Solid-state electronics tend to mess with the sine wave of an AC system. Better LED bulbs have built-in rectifiers, filters, power correctors so that they do not distort the power on the main line. Some have better power correctors than others. This distortion can cause the flicker in some of the bulbs on the circuit. If it is bad enough, the bulb will strobe continuously, even after startup, especially if put inline with a dimmer that also has the characteristics to cause this effect. If it is a minor problem, the flicker stops once the bulbs warm up, get charged, or synchronize, depending on how you want to put it without getting too technical.

The bulbs DB shows may only have a better capacitor, to even out the flicker in individual bulbs that is specifically inherent to the 60Hz cycle, but not have a filter to clean up the power distortion that effects other bulbs. (Just a for instance, not going to look it up) Sometimes, just changing one bulb to a different brand, can stop/disrupt this effect, or putting in one incandescent bulb. Of course, that is not a favorable option, but if you put a standard incandescent in one of the sockets that does not flickers, and it stops the flickering, it may show that the bulbs just were not playing well together.

Or, maybe pull out all the bulbs but the one that flickers, and see if it still flickers.

I bought some LED strips from ReStore and they required 12 or 24 volts DC (some of each). I tried powering them with "electronic transformers" but they didn't work well, but when I purchased a magnetic power supply, they worked fine. I can dim them using a Variac.