Yes, you can. While processor response codes are available by default, additional processor responses are not available to merchants on the web interface unless explicitly enabled. If enabled and the processor declines the transaction, the processor’s raw response will be displayed on the transaction in the Control Panel. 

You can enable it by navigating to 
Settings >> Processing >> Display Additional Processor Response and clicking Enable (see screenshot)

Processing_Options_-_Braintree_Gateway.png