If you mean responsive up to resolution X you won't find many issues, although even then it doesn't make sense to use REMs for borders for example. I can't see any point in using REMs for a border when you've designed it specifically to be 1px. The majority of design elements like that are not meant to scale.
Now try to do a fullscreen responsive site, that can handle up to 2560x1600 and look good across everything. I guarantee you that REMs are not the epic solution they may seem; for typography they're fantastic. But even for padding you're still going to want custom values depending on resolution (since font-size does not necessarily scale linearly with resolution to look good), at which point it makes absolutely no difference whether you use pixels, percentages or either of the EMs
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