I will say yes, but, that answer doesn't necessarily hold true 100% of the time. It really depends on the bands that the device supports.
On the device spec sheets you will usually see something like this:
800/1900/700
These are the bands that the device will operate on. I pulled these numbers from the Droid Bionic. From that spec sheet, it goes even further and shows that it supports EV-DO rev.A (CDMA/3G), and LTE (4G). So this phone does support Verizon's 3G & 4G networks, which means that if 4G is not avaialble, then it will fall back to the CDMA data network (which is what the device will uses for Voice even when data uses LTE.
The same rules apply for GSM networks like AT&T and T-Mobile. The device spec sheet will say what bands the device can operate on, and if the carrier's network operates within that band, then the phone will work. This is why iPhone users can use unlocked AT&T iPhones on T-Mobile's network, but they only can get EDGE data. This is because the iPhone does not support the band that T-mobile uses for its UMTS/HSPA/HSPA+ (3G/4G) data networks.
T-mobile bands:
- 850/1900 (GSM/GPRS/EDGE)
- 1700/2100 (UMTS/HSPA)
AT&T bands:
- 850/1900 (GSM/GPRS/EDGE)
- 580/1900 (UMTS/HSPA) (3G)
- 700 (LTE) (4G)
Verizon bands:
- 800/1900 (CMDA/EV-DO) (3G)
- 700 (LTE) (4G)
Sprint doesn't know if they want to use LTE or WiMax so I am not listing their bands.