Yes, that is a known issue with many Bluetooth headsets, which is not specific to VoiceAttack.
Bluetooth audio uses "profiles", which specify the capabilities of the device, and how data is to be transceived. Manufacturers (either of the device itself, or of the chipset) decide which of these profiles the device supports.
Unfortunately, while devices may support high-quality audio when only the speakers are in use, many devices only support low-quality audio while both the speakers and the microphone are in use.
Essentially, they use a profile that's more suited to phone calls, which uses heavily-compressed audio (basically what isn't necessary to convey the sound of a human voice is discarded).
As mentioned, which profiles are supported is determined by the manufacturer, so unless there happens to be a firmware update for your specific device that adds a profile to provide high-quality audio bidirectionally, that's all the device will be capable of.
Manufacturers generally don't tend to specify which profiles are supported, so often you can't know ahead of time.
Brand, quality, or expense aren't necessarily indicators either.
A (USB) headset is recommended for speech recognition.
If you want something wireless, make sure it explicitly supports PC, and includes a USB dongle. E.G. there are rather expensive Xbox headsets that have Bluetooth, but don't actually support high-quality audio unless you have a specific Xbox accessory that allows the device to connect to your PC wirelessly (which will most likely use a proprietary protocol internally, rather than "proper" Bluetooth)