I suspect my silicon may be too old. I've had the boards for a couple
of years.

The build/date codes on both boards are: QIAAAA/1716AK.
The labels on the jlink chips are:
V0.9.2 2017.34 683955518
V0.9.2 2017.33 683137769

Some of the Nordic examples log errata 104 & 154 as enabled.

Further, the usbd_cdc_acm example from the
14.2.0 SDK works, while the one from 16.0.0 fails to enumerate.

Hi Steve,
There are various issues with your silicon versions:
1) SDK 14.2 was the last version to support engineering A silicon as per https://infocenter.nordicsemi.com/index.jsp?topic=%2Fcomp_matrix_nrf52840%2FCOMP%2Fnrf52840%2FnRF52840_ic_rev_sdk_sd_comp_matrix.html
2) high voltage operation does not work on engineering C silicon or older, errata 140
3) multiple USB erratas, 162, 171, 180, 200, which prevent USB from working

I think I'm wasting everybody's time and should just get a couple of
current boards.

That would be a good idea.
Thanks,
Jamie