@snoobler 

Set the Archer A6 AP to have a static address. 

Another thing to test would be to enable the 2.4 Guest Network on the A6 AP, and move over the modules to that network to see if the 2.4Ghz could be getting overwhelmed.

@Tony Thanks for replying. Already tried the static IP thing.

Note that the "single router everything fine" comment was incorrect.  It just so happened that the router permitted the 28 devices I was pinging to connect and left others out of the loop. I wasn't paying attention to those.

After some back and forth with Tech Support, I was informed these units have a 32 device limit on the 2.4GHz band and 32 on the 5GHz band.  I was trying to connect more than 32 devices (28 of the ESP8266 and a handful of other devices were also set to that SSID).

I like your idea about the guest network.  That's interesting.  If it actually has a separate radio, I suspect it would work, but I suspect it's actually a virtual SSID, and you may hit the device limit when you sum the two devices on the different SSID.

I was also informed that even the most expensive models of TP Link routers have 64 device limits.  My solution is likely to go back to a NetGear Nighthawk.  I've removed the A6s from the network and gone back to the R7000 Nighthawk but with DD-WRT installed. It's handling all devices without issue, but it has a stability problem that I've wrestled with since it was new - the sole purpose for this exercise.  Hoping DD-WRT might solve the stability issue.

I did confirm with Tech support that these can be chained together to get additional devices online.  My issue is I need at least 34 2.4GHz devices on the same router. Yes, I could disassemble and reprogram them to a new SSID, but buying a capable router takes WAY less effort... unless of course I count the time wasted in this exercise.

Lesson learned:  Allways ask the manufacturer about any limits on the number of devices. TP-Link did not publish this anywhere - not in the manual, specs or product information.

As it stands, I'm returning these A6 units.  I'll still consider the TP-LINK line, but I will be widening my search to other manufacturers if DD-WRT doesn't solve the Nighthawk stability issues (don't expect it to... just taking a hit off the hopeium pipe).

Here are the models capable of > 32 devices per band (pasted from Tech Support's email message):

Archer C8(maxium 2.4G:64 and 5G:64)
Archer C9(maxium 2.4G:64 and 5G:64)
TL-WR840N V1, V2, V3(maxium 2.4G:63, no 5G)
Archer C4000(maxium 2.4G:64 and 5G:64)
Archer AX11000(maxium 2.4G:64 and 5G:64)
Archer AX6000(maxium 2.4G:64 and 5G:64)
Archer C5400X(maxium 2.4G:64 and 5G:64)

Currently, other routers can only support maxium 32 clients for 2.4G.

@Tony I didn't test it, but Tech support said that the 32 device limit applies to the total number of devices on the normal and guest networks, e.g., if you have 16 on one, you can only have 16 on the other.  It's clearly just a virtual SSID.

Thanks again for your reply.

Steve