Income requirements for bursaries do vary. Where we are, household income has to be below £80k to be considered.....and bursaries are on a sliding scale so if you have over £70k I'd guess you'd only get a small amount if help, but I think they say that an income if below £25k could qualify for full help.
Believe it or not, there are some schools who will consider bursaries on incomes up to £120k - I think SPGS is one of them. This is unusual though and again, the bursary offered at that level would presumably be low.
In my experience, bursaries, especially at the higher levels are fairly rigorously tested - there's a lot of paperwork and evidence to provide and so whilst some people might cheat the system, the system is designed to genuinely offer to those with more limited means. Bursary policies normally say that certain things are not appropriate for those receiving bursaries and then list things such as owning a second home, regular new cars, significant home improvements etc. The point is that bursaries are not there to fund the fees of those whose lifestyle choices mean they can't afford fees. Those receiving bursaries are expected to have made sacrifices or that there aren't further sacrifices for them to make - so it is expected that unless there are small children or some other pressing need, parents will work full time and they will liquidate any excessive equity in property and not be engaging in large soending. Most fee paying parents make sacrifices for their children to attend, so bursaries are not there to mean the recipients can continue their expensive lifestyles which leave them with little cash.
The reality is that in most places bursary finds are limited. Bursaries are often offered to those in need who are ALSO very high performers. There will always be children who apply who are bright who might be offered a bursary which is too small to allow them to actually attend. Schools seem to vary in their honesty about what is likely or achievable at different income levels which often causes frustration when hopes are raised and then disappointed. I guess that every year is slightly different in terms of who applies so it is hard to be precise about how much will be offered, but often the amounts seem radically below what people hoped for or felt led to believe they would be offered. Perhaps schools hope that once people have invested the time and effort in going through the draining application and bursary process they will scrape together the extra required, or they simply make their offers knowing some won't be able to afford the fees wih the bursary offer made and that because their funds are limited and/or because they have other fantastic candidates who might be even better, they are willing to let some children go as part of the process. I agree that it's hard if you work hard, receive an offer and are a top performer and then knowing the income and what a family need to send their child, the offer is significantly too low. Really hard if the alternative is a very poor state school......so good to hear in Mrs Patmore's case (and I like to think of Mrs Patmore's from Downton having a child who received a bursary or scholarship to a top public school) that the alternative was a SS Grammar - still so highly sought after and brilliant, even if not quite the unique experience of Westminster. I'm sure he will have gained things by being in the state sector he would never have accessed at Westminster too.