LSE is more competitive for quite a few courses statistically, and since they don't have all the information that Oxbridge do (e.g. admissions assessments and interviews etc), they rely on just a couple of things (like the personal statement) which can lead to one weak area being magnified for them whereas for Oxbridge it might not be such an issue. So it certainly happens.
Also obviously for medicine, all medical schools have very different requirements and someone applying without paying attention to those can mean they are competitive for one of Oxford or Cambridge, but not for others - and thus get an offer from Oxbridge but not other medical schools.
Plus for architecture and fine art where the portfolio plays a significant part, some strong courses in those areas may not respond as well to a given portfolio as Oxbridge do.