For consulting, the cover letter is looked at closely as there are only a few internship positions, and recruiters like to see students who have spoken with people at the firm and done their research as to how each of the MBB is differentiated.
95% of cover letters from MBB are generic and are recycled for each firm, except for a couple sentences changing the title, firm name, and a few buzz words each consultancy uses.
In our experience at MBB reviewing resumes from target schools, the cover letters make a difference for students who are not at the top of the pack of resumes. Those students who have 3.9+ GPAs and stellar internships or campus experiences will most likely get the interview slots, unless a glaring error or negligible effort was put into the cover letter.
Of the many students who are a cut below, the cover letter is often a deciding factor. As an example, an outstanding cover letter for an above-average (but not top) applicant mentioned conversations the applicant had with consultants at the firm, specific examples of interaction he had with their people, and the industry and capability areas where that firm was leader. In this case, the cover letter made the applicant a "second priority" to a "priority" for the interview.