Enter the required navigation performance values as limits and the aircraft will monitor the lateral deviation for you, the alert is Unable Required Navigation Performance. For vertical navigation it is a bit less obvious, but the VNAV deviation scale on the ND will reflect the entered lower value and turn yellow if you leave the entered range. I honestly do not know if you will get the same failure message as i haven't seen it yet for vertical navigation, but have seen it numerous times for lateral, especially after entering an RNP of 0.01NM as the usual ANP is 0.02. On IAN capable aircraft you will get in any case the "GLIDESLOPE GLIDESLOPE" warning message if you deviate too far from the FMC generated glide path, no matter if it is flown in LNAV/VNAV or APProach mode.
In my operation we have to check if the relevant RNP (0.3NM for most approaches) is either automatically or manually entered, otherwise we are not allowed to fly the approach. As we use IAN we do not need to enter the vertical RNP value for normal RNAV approaches, the display is the same as for an ILS and the computer generated glide path follows the same logic. For RNAV RNP AR approaches we enter 125ft vertical RNP and fly it in LNAV/VNAV (the only approach we fly in that mode).