It depends on the machine.
In general, the answer is no. Address decoding is a hardware function and a given address presented on the address bus will be served either by ROM or RAM, but not both. (Another possibility is that there is nothing that serves the decoded address, in which case the data read is undefined.)
When chipset vendors started implementing "shadow ROM" that changed. The chipset was given the ability to copy the ROM areas into RAM and then change the address decoding so that a given address presented on the bus hit the RAM copy instead of the ROM copy. This was also probably combined with a write protect, as the intent was to take advantage of the faster RAM speed.
That was a cheap and easy way to make full use of the RAM present on the system while dealing with the ROM and adapter address area between 640K and 1MB. The elegant solution would have been to shift that overlapping RAM up 384KB in the address space, but that would require additional hardware and have performance impacts. The shadow ROM was a more "hackish" solution that did not allow you use make full use of that 384KB of RAM, but it did provide some performance benefit. Which became obsolete as soon as operating systems stopped using the machine BIOS to do anything, but that's a different problem ...
If you can figure out how your chipset handles the shadow ROM function you might be able to do it. There might be a write protect register that you can find that would allow you to change the contents of the RAM after the ROM copy is done.