FWIW, In the NWS, I'm hearing that PhDs are popping up in Met-Intern applicant pools more often. (Met Interns being the entry-level Met position in the NWS, not an actual "internship"). So, yeah, the competition is becoming fierce!
My advice to anyone trying to get into the NWS after Undergrad: Try to get into the Pathways program (formerly SCEP). That way, you're virtually guaranteed an Intern position assuming you don't screw up and aren't picky about where you end up. Hiring overall has been at a glacial-pace despite hundreds of vacant positions, so I'm not sure when the next time these Pathways positions will be bid out. We're only just now bringing in students that applied roughly a year ago. Rumor has it that some students are even delaying graduation so they can still qualify.
Otherwise, try and get the most operationally-relevant Summer Internship possible. Met Intern applications are initially scored based on how you answer a bunch of experience questions. The more NWS or operationally-relevant experience you have, the better off you are. When applying from outside the federal government, only the Top 3 or so applicants are forwarded to the Selecting Official (MIC). If you make it past that round, that's when the "stand out" experience really helps. GIS is good to have (and is relevant outside the NWS). Skills with graphics, briefings, video software are becoming more useful every day. "Decision Support Service" is the focus behind "Weather Ready Nation", and basically means working closely with Emergency Managers and other decision-makers, and trying to speak their language. If you can land an Emergency Management-related internship, that will stand out to an MIC, even if meteorology isn't the focus. But it might not help as much with getting "points" on the initial Application unless they start changing the questions to reflect the DSS-focus. So try and be well-rounded. Do the EM and graphics stuff, but not at the expense of the meteorology. Having an advanced degree gets you big points on the initial application too. Veterans who qualify for the job float right to the top of the list.
All that being said, if I were an MIC, I'd be biased towards hiring "Weather Geeks" that have the extra skills. The reason is simple: Weather geeks will be self-motivated and their morale won't be as affected by the other "distractions" inherent in government work (bureaucracy, Congressional funding games, etc) which tends to wear some people down over time.
I hope this is helpful to someone out there slaving away at thermodynamics homework on a Sunday afternoon.