If you're using pixels as part of your responsive designs you're probably making your life harder than it needs to be.
There's nothing "wrong" with using pixels in an otherwise responsive layout, but if you do you'll likely end up writing more code than you would using flexible units.
Jon Allsopp's A Dao of Web Design predates responsive design by a decade, but its prescient advice remains perhaps the best way to approach any design, responsive or otherwise: "It is the nature of the web to be flexible, and it should be our role as designers and developers to embrace this flexibility, and produce pages which, by being flexible, are accessible to all."
More than just embracing the nature of the medium, building your sites atop what developer Trent Walton calls "Flexible Foundations" can go a long way to making development easier. As Walton points out in his post, using pixels often means more code since pixel-based type, margins and padding mean you need to add new values at every responsive breakpoint.
"In many ways," writes Walton, "every time we use a pixel value in CSS we're rasterizing what was a fully-scalable web."
Stick with percentages, ems or the newer rem units and your designs can scale simply by changing the body font size. Embracing the flexibility of the medium means you can adapt as well – no need to panic when a client wants to make the logo bigger at the last minute, you can scale the whole layout up (or down) with a few lines of code. For Walton's design firm, Paravel, the flexible approach has already proved its worth in just this way:
As developer Brad Frost has said, "Get your content ready to go anywhere because it’s going to go everywhere." Pixels may work today, but they make for a rigid site that might well break on new devices. As Walton concludes "the sites we've built to display on a desktop, smartphone, or a tablet today could be on a TV screen, coffee table display, or mega space yacht projection system tomorrow." Start with a flexible foundation and your site should handle just about any hardware that tries to load it.