I think the limiting factor here is the Bands / frequencies that are (and aren’t) supported on the E371 module.
I found this on the E371 spec sheet:
LTE AWS, 700 MHz (B17) (with MIMO)
HSPA+/UMTS 850/AWS/1900/2100 MHz
GPRS 850/900/1800/1900 MHz
Elsewhere, I found information that says T-Mobile uses Bands 2, 4, and 12 for 4G LTE. This tells me that 4G LTE via T-Mobile is not possible with this module (with this Digi product), because only Band 17 is supported, which I believe is the main band for AT&T 4G LTE.
So, unless I’m missing something, that information alone limits you from using 4G LTE via T-Mobile.
The same question/logic would apply to 3G. From the quick research I did, these are the supported 3G bands (well, frequencies) for T-Mobile:
1700 MHz, 1900 MHz, and 2100 MHz.
So maybe 3G would work in certain areas(?).
At any rate, this Digi device is definitely marketed and intended for use with AT&T only, and likely only supported when used with AT&T (just so you know).
If you have a working T-Mobile SIM card, you could certainly try it. Just be sure to use the proper APN for that SIM.
Sorry I don’t have a definitive answer. This isn’t something I’ve tested before. I hope this information helps still, if even somewhat.