If you were a well-off merchant, could you bribe your way into knight-hood/lower tier nobility complete with a grant of lands? Or was that rare, something that required unusual political circumstances, or both?

Yeah, you could, and you can find people complaining about up-jumped merchants buying their way into the gentry and the squirarchy and then into the ranks of knighthood as the 12th and 13th century. 

But as one professor whose lectures I listened to described it, it wasn’t like medieval England was marked by huge amounts of upward social mobility. Rather, what you had were spurts of ambitious social climbers and monarchs/nobles in need of cash, and then a good deal of social anxiety about what was happening to the old noble houses, and then people would redraw the boundaries, and there would be a pause – and then you’d have another spurt. 

The irony was that, because these social climbers were desperate to fit in, they often ditched any signs of their bourgeois pasts (including getting out of trade and into land) and fully assimilated into the value structure of the nobility, so that after a few generations (especially with intermarriage) you really couldn’t tell them apart. In this fashion, the professor argued, what you had was a ruling class that refreshed itself every so often with new blood but with very little change in ideas or social structure, and a very small amount of upward social mobility into the ruling class but a good deal of fluidity within the ruling class.