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on the 'old' system, which we have currently, you have a breaker on each circuit (which protects against too much current being drawn). The two RCDs are then protecting half of the circuits each, so if there is a fault where more current goes out than back in on any of those circuits the RCD trips. Normally you get it laid out something like this in the consumer unit: RCD..[a set of breakers] RCD [a set of breakers]
This means if the RCD trips all the circuits connected off it wont work. The 'new' style is to have the RCD and breaker combined for each circuit (RCBO) so only one circuit trips for a fault.
It's not absolutely necessary to pass an EICR, you can just upgrade the two RCDs (if there's space in the consumer unit) if they're not the type that is now required.
An RCD tripping will take out any circuits connected to it, RCBO will only trip the circuit causing the issue.