I'm reasonably competent, so I can recognize a warning that refers to a line of code I've just changed, even if there are other warnings.
My counterargument would be that I think this sounds very error-prone. I don't think it's a matter of "competence", I think it's a matter of "I just changed a header and now it's rebuilding 100 files and so noticing a new warning requires scanning thousands of lines of output."
The one thing I'll grant is that there's an opposite direction you can go. Instead of disabling warnings you don't like, if you -Werror those that you [i]do[/i] like, that's also sufficient.
Let me ask this: do you have on all warnings when you build? Unless, on GCC for C++, you're passing -Wall -Wextra -Wabi -Wctor-dtor-privacy -Wnon-virtual-dtor -Weffc++ -Wstrict-null-sentinel -Wold-style-cast -Woverloaded-virtual -Wno-pmf-conversions -Wsign-promo -Wformat=2 -Winit-self -Wmissing-include-dirs -Wswitch-default -Wswitch-enum -Wsync-nand -Wtrigraphs -Wunused -Wstrict-overflow=5 -Wfloat-equal -Wundef -Wshadow -Wunsafe-loop-optimizations -Wcast-qual -Wcast-align -Wconversion -Wlogical-op -Waggregate-return -Wno-attributes -Wmissing-declarations -Wmissing-noreturn -Wmissing-format-attribute -WWnormalized=nfkc -Wpacked -Wpadded -Wredundant-decls -Wunreachable-code -Winline -Wno-invalid-offsetof -Winvalid-pch-Wdisabled-optimization -pedantic? (Possibly there are a couple redundant ones there, but GCC has lots of warnings included in neither -Wall nor -Wextra.)
If you're not passing those to GCC, what that means is that you have, implicitly or not, decided that the cost-benefit ratio of the warnings you are not enabling is not in favor of enabling them. Well, saying "there are too many of these warnings, I'll disable them" is making the exact same decision: that the cost-benefit ratio of leaving that warning enabled is not in your favor.