Here are a few typical hardware problems you may encounter.
Check your Power Supply Unit (PSU), if it is a 12V PSU, the regulator will likely become too hot and fail to generate 5V successfully. Switch the PSU for a 9V PSU (one supporting a 2A current will work with both an Atmos and a Cumulus plugged).
If the Oric displays the cursor at the bottom right of the screen AND many vertical lines (1 pixel large), your VIA 6522 is probably faulty. You should change it.
Be careful if you have a 65C02 with the connection between the 2 cpu pins on your motherboard. If you plug a 6502 after this modification your Atmos should refuse to boot.
If you have access to an EPROM writer, FF (who?) have written a diagnostic ROM which can help determine if and which RAM chips are faulty.
You can download it at http://oric.free.fr/ROMS/diag.rom
FF have not published usage instructions for it yet, but you should be able to understand what it says (yes, it talks, but in french. FF would be happy it helps you determine what is wrong, please try it : it can run even if all the memory chips are faulty (this means that the code only use the CPU registers, it cannot rely on any ram byte, thus it does not use subroutines (because the stack is in memory), nor pointers (zero page), etc.)
If a single RAM chip is faulty, its bit number will be displayed on the 3 last lines of screen. If several chips are faulty, a pattern will be instead displayed on these 3 last lines pointing out which chips are bad. If your CPU, VIA and PSG are ok, the will also announce the bits of each faulty RAM chip on the loudspeaker.
Note: the French words you will hear if chips 1 and 2 are faulty will be :
zero : oui ! un : non ! deux : non ! trois : oui ! quatre : oui ! cinq : oui ! six : oui ! sept : oui !