I found on various sites that you can Dremel the end of a x1 PCIe slot to fit a larger (x4, x8 or x16) PCIe card. However in my case, my motherboard was still under warranty, so I didn't want to alter it. Also, the motherboard had various components in-line with the x1 PCIe slot that would have prevented a larger from fitting. Fortunately, I had a surplus of older x16 PCIe graphics cards so I decided to cut the card instead.
Armed with the knowledge that the card worked properly on an x1 bus, I set out to modify the card to fit.
Before I continue, I must point out that I'm not recommending anyone do what I did. Attempting to do the same modifications will definitely void your warranty and the process will expose you to respiratory dangers (a respirator might be sufficient, but I'm not sure as the dust is exceptionally fine). Again, if you decide to do something like this, it is at your own risk.
First, I used a permanent marker to mark the first pin that needed to be removed. According the pinout, PCIe x1 only uses the 7 pins after the notch. So, I marked and then cut the 8th pin. I aired on the side of caution, cutting closer to pin 9 than pin 7. Since I was using a grinding disc, I was then able to grind the PCB down a bit further toward pin 7.
Since the motherboard I was using had integrated components in-line with x1 PCIe slots, I had to remove the remainder of the pins not used by a x1 PCIe card/slot. So, starting from the back of the card, I removed the rear hook and the unneeded pins. The final result looked, not surprisingly, very much like a standard x1 PCIe card.
The whole point of this exercise was to add a second PCIe graphics card to a motherboard that had only one x16 PCIe slot and two x1 PCIe slots. In this scenario, the x1 PCIe card would obviously exhibit very poor gaming performance. Fortunately, I only wanted an additional graphics card so that I could add additional monitors for general productivity
I installed both a stock, x16, version of the graphics card and a 'neutered', x1, version of the graphics card. I booted Windows and waited for it to Plug-and-Play the new cards. After a reboot, both cards were working without issue in Windows.
It is debatable if the risk to the card/your health is worth the modification. But, at least from a technical standpoint, it is possible to install a x16 card in an x1 PCIe slot.
Addendum: In my research, I also came across the YouTube video below. It shows someone performing the same procedures. As I mentioned earlier, note that they are wearing a respirator while cutting the card. Exposure to dust created from grinding the PCB/components could present serious health concerns if inhaled.