The anecdote that comes to mind for this phenomenon is the more frequent flyer miles you have the more like you are to get a negative experience such as being bumped from a flight. This is because they know you're invested in their airline and are more likely to come back. A new customer might decide to take a different airline next time.
Edit: Apparently this was an old wive's tail or something. Oh well...
With a lot of frequent flier miles you are very likely a status customer and airlines will bump those last and, depending on the status, will bump other low revenue passengers to make space for a high status customer.
I know a bit about aviation and if an airline really needs to involuntarily bump a passenger (which is quite rare) they will chose an infrequent flier with a cheap ticket.
Unless you can provide a source for your assessment I call bullshit.
This is accurate per everyone I know at airlines. The reason is it’s trivial to shift one’s loyalty. If an airline goes to shit, I’ll switch from spending cash with them to burning down miles while I book work trips with someone else.
There are two classes of frequent flier miles: status-qualifying miles and award miles.
Status-qualifying miles are miles that you can only earn by flying with that particular airline. Those miles are used in calculating your "status" with the airline. Achieving a higher status generally comes with perks that make frequent travel easier and more enjoyable (such as being able to sit anywhere in the plane for free, extra checked bags, access to lounges and better chances at upgrades).
Award qualifying miles are the miles that people most commonly associate with frequent flier miles. These are the miles that can be redeemed for discounted flights in the future. There are several ways to earn this aside from flying with the airline (though flying with the airline gives you more award miles than not): credit card spend, promotions through airline partners, giveaways, etc.
There is also another metric that airlines collect to determine loyalty: the amount of money you spend on your tickets, or status-qualifying spend (or dollars). In the past, this information was only used to protect more frequent or high-paying customers from being inconvenienced during travel. Now, airlines use it as an extra requirement for achieving status levels.
Now, since award miles for airlines can be collected without flying with them, flyers with very little status-qualifying spend will usually be the first in line for involuntary removal (involuntary denied boarding, or IDB). However, IDBs have become less frequent ever since the United Express incident last year; airlines are trying to favor giving people vouchers first (voluntary denied boarding, or VDB) before bumping folks.
That doesn't match my experience. When I have a large reserve of frequent flyer miles I am generally treated quite well, actually. Frequently I have been bumped to first class gratis, or "economy plus" if that is what is available, given discounts/rebates on luggage and other fees, offered complementary access to the airline's lounge/club at the airport, etc. This is all above and in addition to the usual perks, like priority/VIP boarding order or whatever.
"Frequent flier miles" don't mean a thing on any major US airlines. The only factor in how they treat you is your overall profitability, which is calculated in elite qualifying miles (EQMs). This is what determines your status and your status determines your perks. These miles come from buying high cost-per-mile tickets, i.e. a $300 SFO-LAX fare, a $20K international first class R/T from IAH to LHR, etc.
Edit: Apparently this was an old wive's tail or something. Oh well...