Hardware
a.) If you have installed a new graphics card, have you updated your bios, and checked that the PSU is sufficient.
b.) C3 is very 'good' at detecting unstable ram, if lowering the speed/tightness of the timings improves stability you may need to read up on getting the ram stable on your platform.
c.) If unstable on an overclocked setup, have you tried playing with default settings to see if its the system stability at fault.
d.) Crossfire and SLI setups increase the requirements for ram and cpu, have you tried turning one card off and checking stability. Does your PSU have sufficient power?
e) Is your power supply sufficient? I am putting this point again because so many people have upgraded old computers with new ram, and new graphics cards but are using borderline PSUs, especially a problem if they are old. Very old PSUs (especially/only if they are not good quality) can fall below rated capacity after a few years.
Graphics Drivers
a) Are you using the latest drivers for your graphics cards?
b) Have you done a proper full uninstall reinstall of the drivers to ensure they install correctly? (especially AMD beta drivers). If problems persist please read up on the interweb on how to do a complete uninstall of the graphics drivers, its somewhat advanced and I don't want to be blamed for people making mistakes and breaking their registry.
c) Check your vendor for vendor specific drivers, the globally validated drivers do not always allow full stability of vendor specific cards (i'm looking at you HIS and powercolor), especially if the card is overclocked and are not reference cards.
d) If the vendor drivers are out of date (often), you can try installing the latest vendor drivers and then updating over the top with the latest AMD/Nvidia release. This was often required in the C1 days. Its still sometimes required today, the install process for the graphics drivers on my Gigabyte 7970s OC produce an error unless I install over the top of the vendor drivers.
c) If you don't have a high refresh rate monitor (eg refresh rate greater than 60hz), you can try limiting FPS either by cvar or on the driver side to 60 fps (if driver side some people recommend 61-63).
Nvidia specific
The Low texture levels and very high texture levels settings are causing some issues on some nvidia cards, try modifying this graphics setting. There have been some reports of the 3D drivers causing instability.
AMD specific
If upgrading to crossfire, you have to uninstall the graphics drivers first, then install the new card, then install the new graphics drivers. Otherwise there could be instability problems. Secondly lots of people have reported instability problems with installing two bridges on a 2 card cf setup. Lastly, if more than two cards.... may god have mercy on your soul.
Cryengine related
a) Try changing graphics settings, the medium/high global settings are the most stable from what people have reported. You can also try the cvar configurator http://secure.gamesas.com/forums/viewtopic.php?f=65&t=56955&hilit=support+thread
There are some great tips and some specific trouble shooting advice in that thread. Check it out if you have tried tweaking global or individual settings within c3 to no avail.
b) Avoid very high if you are using older gen AMD and Nvidia cards, the tesselation is a bit bugged and can cause issues. There have been reports of major issues with tesselation enabled on 5000 series AMD and 200 series Nvidia cards.
c)Edit: Vsync is buggy. Try playing with it off, if you can.
NAT
Theres a ton of ports that are on the website/manual that you need to forward/trigger/etc to prevent nat issues. If having multiplayer connection issues, try unblocking the ports... It will depend on your router/firewall.
b) In Zonealarm, for the first time I had to manually go in and create rules for crysis 3 that specifically opened the designated ports. For some reason auto mode full permission was insufficient.