Full disclosure: I work at Braintree. If you have any further questions, feel free to contact support.
There are a few known issues with the Magento2 + Braintree module that I'm aware of, but I'd like to clear a few things up with you first.
First and foremost, the test credit card 4111111111111111 will only successfully create transactions when your Magento site is connected to a Braintree sandbox account. If your production account is connected, then it will still create a transaction/verification however it will do so unsuccessfully and with the status "Processor Declined".
As long as your sandbox API keys are connected to your Magento2 store, some general issues others have noted in the past are:
The input for descriptors in your Magento2 store do themselves contain validations that Braintree's API has on the fields when they're passed in a Transaction.Sale() API call. As such, if you have any descriptors in place and they do not adhere to the requirements listed in our developer documentation, that could case this issue.
This would manifest itself as a 400 error on your page in the network tab so it's unlikely you're seeing this, but there was a known issue where the state/region portion of the checkout page would cause a 400 error to be passed back. If it happens it should appear in the payment-information network request during checkout
That said, if neither of those things are related to your issue please feel free to write in to our support team and we can verify your Braintree account settings and see if any API activity is making it to our servers. It might also be worth it to check your system and debug logs in
/var/log. You can enable the debug logs for your Braintree module by navigating to Stores > Configuration > Sales > Payment Methods > Configure (Under Braintree) > Advanced Braintree Settings
and set Debug = Yes.
Finally as Keyur Shah mentioned in his comment, the magento2 github issue section is fairly well fleshed out with reported issues/solutions so I'd definitly check there as well!