Bluetooth (BR/EDR) and Bluetooth Low Energy (BLE) are pervasive wireless technologies specified in the Bluetooth standard. The standard includes key negotiation protocols used to generate long-term keys (during pairing) and session keys (during secure connection establishment). In this work, we demonstrate that the key negotiation protocols of Bluetooth and BLE are vulnerable to standard-compliant entropy downgrade attacks. In particular, we show how an attacker can downgrade the entropy of any Bluetooth session key to 1 byte, and of any BLE long-term key and session key to 7 bytes. Such low entropy values enable the attacker to brute-force Bluetooth long-term keys and BLE long-term and session keys, and to break all the security guarantees promised by Bluetooth and BLE. As a result of our attacks, an attacker can decrypt all the ciphertext and inject valid ciphertext in any Bluetooth and BLE network. Our key negotiation downgrade attacks are conducted remotely, do not require access to the victims’ devices, and are stealthy to the victims. As the attacks are standard-compliant, they are effective regardless of the usage of the strongest Bluetooth and BLE security modes (including Secure Connections), the Bluetooth version, and the implementation details of the devices used by the victims. We successfully attack 38 Bluetooth devices (32 unique Bluetooth chips) and 19 BLE devices from different vendors, using all the major versions of the Bluetooth standard. Finally, we present effective legacy compliant and non-legacy compliant countermeasures to mitigate our key negotiation downgrade attacks.