One of the most common (and frustrating) aspects of the job search process for both me and my candidates is waiting to hear back from firms after a submission or an interview (*especially* an interview). While the information and advice that follows won't necessarily speed up the process of waiting, hopefully knowing the dynamics will make the wait a little easier or less stressful.
The main reason you haven't heard anything right away may seem obvious, but bears repeating because it is often forgotten from the standpoint of the candidate - everyone is very busy!
The recruiting coordinator at the law firm who receives and screens the initial applications is busy - in the current market, it is not uncommon for any given job posting to receive dozens, if not hundreds, of applicants, especially in the larger markets. During certain times of the year, recruiting coordinators may also be inundated with the work surrounding on-campus interviews, coordinating the orientation of new hires, preparing for and networking at industry conventions, etc. They work extremely hard to keep on top of everything, but this often means that it can take a long time for even the most diligent and responsive recruiting coordinator to be in a position to respond to a submission.
The hiring partner(s) or other attorneys are very busy as well (or at least you should hope so if you are applying to the firm and want to be somewhere with sustainable work levels). If there is a pressing client deadline, you can bet that reviewing the resume and cover letter of a potential candidate, while important, is nonetheless lower on the to-do list.
Finally, everyone whose input is necessary for the decision to reject, interview, or make an offer to a candidate needs to be on the same page before a communication is made, so it can result in a long (sometimes unbearably long if you are the candidate or recruiter) wait before you hear anything. This isn't even taking into account things like conflicts checks, reference checks, the fact that firms will almost always interview multiple candidates for any given position (which can push the process from weeks to months depending on scheduling difficulties).
In short, silence does not necessarily mean something negative, and it does not mean you are a bad candidate. I recently received an interview request for one of my candidates over two months after the candidate was submitted for an open position, and without receiving any communication from the firm in the interim (despite a couple of follow-up emails). A fellow recruiter mentioned a candidate the other day who received an offer four months after interviewing with a particular firm, and had assumed that the position had fallen through in the meantime.
The best thing to do is to follow up periodically (I usually check in every two weeks or so - not too frequently so as to be a pain, but frequently enough so that my candidates do not fall through the cracks) with a polite email indicating you are still interested in the position, and offering to provide any more information that might be helpful to the firm in making their decision. Waiting is no fun, and from an outsider perspective it can appear as if the firm is just being non-responsive, but this is generally not the case - firms take their hiring very seriously, and take applications for open positions very seriously, but for all the reasons outlined above (and a few others), the process can be slow. Hopefully knowing what is going on behind the scenes, and being aware of the most likely reasons for the wait (don't assume the worst) will make the wait more bearable. In the meantime, have patience, stay positive, and be sure to proactively and politely check in from time - it never hurts!