Point taken about radar envelope, if you are talking 1006-1008 India band navigational radar. Bear in mind though, the job of these warships is not only to prosecute a contact but to detect it, the radar envelopes of eg. 1022 surveillance radar are quite massive and will detect both terrestrial and airborne craft. Additionally, there are ways and there are ways of detecting anything transmitting radio waves, without the need for radar. Passive detection will yield almost staggering accuracy at considerable range, ESM. Your jet will not get within X hundred miles of a FF/DD without being 'seen', either by radar or otherwise. If indeed your aircraft is heading towards the warship under the Vertical Coverage Diagram (envelope) then you will certainly be contacted as the warship will start to twitch...!
Nuff about boring tech stuff, you are correct that nothing in this life is failsafe, shooting down aeroplanes is a US Navy stereotype, crashing into land is most definately our forte, can name another half dozen serious incidents in last 2 years that you won't have seen on the news!