@megan
Yes they can, because I've had it. Trouble is, I can't recall the root cause (we're talking 20 or more years ago), but the regulator was wondering if the aircraft should be certified IFR capable with such a single point failure. Nothing eventuated. Contact with Australian ATSB or the CASA Moorabbin office should elicit a copy of the incident report.
I did that already 2008. I got the incident report and other related material from Australian ATSB.
Think about it? Is this event to the lose of Dual AP's a root cause or is this the consequence? For me it was very clear after receiving these Australina ATSB's document which was verified and evaluated together with the other non-public documents which were on my hands.
This is a known issue in general but the all related documentations are non-public and classify. You can wonder why?