The answer is, it depends.
The history of 3D printing includes a time when engineers turned away from using these devices because they could not deliver a quality built part. Vendors went back to the drawing board and designed new printers, using advanced technology that came about during that time period, that could build objects to meet engineers’ form, fit and function needs. But there was a trade-off. To ensure a good build, the machines were designed to work with materials with specific characteristics, such as flow rate, melt rate, size, and so on. Vary those parameters and you risk not getting a quality part. This is part of the reason vendors recommend specific materials—they know these materials will not gunk up the machines because they’ve tested them.
The requirement for the use of specific materials is easing, especially for machines that use powders. As technology advances and vendors can develop more capable machines at a reasonable cost, the range of usable materials will expand.
If you have a maintenance and service contract, check with your vendor before using a non-recommended material. You could nullify the contract.
It’s always a good idea to check with the vendor anyway. Many powder-based machines can handle other brands of powder as long as the material meets certain size and shape characteristics. And, often, the vendors will work with you on that.
The other side of this issue; many users want to create their own proprietary material for a printer for specific customers. In these cases, vendors can work with you to develop that custom material.