In my experience, human resource departments are among the most bureaucratic places in a company, and much less enthusiastic and responsive than many others. Employment is one of the more regulated areas in a company so that not all the blame falls directly on the HR workers, but it also appears to me that it attracts people who thrive in these conditions. Be that as it may, what I have observed is that they are often slow.
Two very similar anecdotes, one personally and one first-hand account, may illustrate that: Engineering departments in two different companies (many hundred and many thousand employees, respectively) had their dream candidate to fill a position — in one case actually a very capable intern. The human resources department was informed about the candidate, everything they needed served to them on a silver platter, and still they needed 6 or 8 weeks to get the offer out to the respective candidate, by which time they had of course found something nice elsewhere.
So slow reactions by HR departments are, in my experience, common to the degree that they may impede the hiring process, even for regular engineers. If you feel it's starting to threaten the internship I think it's OK to send a friendly short email to one of the people who you'll do work for and tell them you are still waiting for an answer from them even though time is short to finalize things now. HR may simply need a little reminder to get your unimportant joblow priority task bumped to the top among the many other things that are on their plate.