A key challenge in realizing simultaneous radio transmission and reception is suppressing the so-called self-interference (SI) caused by coupling of the own transmitter (TX) signal to the receiver (RX). Moreover, the inherent nonlinearities of the TX and RX front-end components can seriously limit the achievable SI cancellation. In this paper, we present an active radio frequency (RF) cancellation architecture for SI suppression in radio transceivers operating under nonlinear TX power amplifier (PA) and the RX low noise amplifier (LNA). The proposed technique is based on pre-distorting the TX signal to reduce the unwanted PA-induced emissions, and then creating an opposite-phase baseband replica of the linear SI in the transceiver digital front-end through adaptive filtering of the known transmit data. The equivalent RF cancellation signal is finally generated in an auxiliary TX chain and combined with the received signal at the RX LNA input. A closed-loop parameter learning technique, based on the decorrelation principle, is also developed to efficiently estimate both the PA pre-distorter and the digital cancellation filter coefficients in a flexible manner. Experimental results show that the proposed scheme can achieve up to 50 dB SI suppression at the RX LNA input, even at high TX output power and with wide transmission bandwidth, thereby enabling improved TX-RX isolation while reducing the linearity requirements of RF components in in-band full duplex (IBFD) radios.