No, it's not all about Oxbridge but if you're not in say the top 10-15 unis then you will struggle to get interviews at the top strategy houses. Likes of Accenture, PA etc. take from a wider selection, but still has to be fairly reputable. You should ideally have 2+ short internships or volunteer initiatives on your application (or 1 if it was a full year placement), most applicants do and competition is fierce. Outstanding candidates will get say about a 70% hit rate on being invited to interview (contrary to belief, they do get turned down quite often for being too cocky), most reasonable candidates will get about 30% hit rate. From there, it's how you do on the day(s) that determine if you get the job.