If I recall correctly, the Stark family has had to put the Boltons in their place more than once in their long history.
Even family ties of blood don't mean absolute loyalty, as we have seen with the Karstarks in the last few books. But other houses might be loyal out of some great favor done in the past, such as the Manderly's absolute loyalty to the Starks.
As for could one of the great houses crush a rebellious bannerman, the answer is yes and no. A great house is the way it is because throughout history it has managed to maneuver the halls of power adeptly - both in alliances forged and enemies chosen, and in terms of favors owed, favors gained, charm and intimidation. It is not a simple question to answer.
And... it is because of that tangled web and lack of simplicity that king's throughout history have manipulated their own bannermen, ie- the great houses. A great house's bannermen will likely jump at the chance to help their liege lord crush another bannerman. Why? In addition to the obvious riches to be gained in raiding and looting, there are lands and grants and titles and other possessions to be gained. The less obvious are any perceived slights or debts owed or to be repaid from past history - nothing quite like destroying an old rival house when that war has been sanctioned by your liege lord. Think about that for a moment. As much as the great houses play the Game Of Thrones against each other, so do all their bannermen at the "next level down".
In my own campaign, the player's house has allies and enemies amongst the other houses as at their level. One enemy is an old one, and one they would dearly welcome the chance to wipe from the face of Westeros forever. Another enemy is simply that because that enemy is an serious enemy of one of their allies. And their third enemy house isn't so much a hated enemy, but more of a rival... someone to compete with... they wouldn't save each other, but neither would they go out of their way to cause harm.... I populated my player's region with 12 other houses with which to work - 3 enemies, 3 allies, and the rest all possibilities.... It allows my players to fully engage in The Game Of Thrones. All of it with bannermen at the ready.