Yes, it is possible. Windows is just software. Software is a series of instructions for a computer to follow.
Think about another type of a series of instructions: a book. What can those instructions accomplish if they are written in a book that sits on a shelf and nobody bothers to open the book and read those instructions?
Just as those written instructions require a person to read the instructions and start following the instructions, computer software requires hardware to do stuff to be useful. Even if a book has instructions which were written with fabulous accuracy, that doesn't prevent problems if a person decides to read the instructions but then to implement them wrong. Similarly, software cannot prevent hardware from doing bad things. So, broken hardware can physically triumph over what any piece of software can do, including Microsoft Windows.
Now, ReFS may be designed with the intent that software will store details about the data, and to have the software compare those details later. A simple concept is "checksum", where software adds certain values and makes sure that those values match an expected result. When hardware implements that software, then certain bad results may be able to be detected. This may even be highly probable to work. However, since the number of potential problems, that might theoretically exist, is basically an infinite number, there is no guarantee that software will necessarily detect every single problem. (Keep in mind that software is a series of instructions that was created ahead of time.)
FAT is particularly low on features. FAT12 was designed for floppy disks, and FAT16 for systems up to 4GB (although most of Microsoft's implementation of FAT16 tended to not work above 2GB). Without the VFAT extension, neither of them supported filenames longer than 11 characters (some of which would be in a portion called the "extension"). FAT was simply designed to store data in a time when the ability to store data was a novel concept that adults needed to be taught about. When FAT was considered a "leading edge" technology, computer technology was not yet sufficiently prevalent and elaborate for people to be worrying about advanced features.
NTFS added support for some more features, perhaps most notably having the operating system be able to easily keep track of user permissions. There are different versions of NTFS. For instance, Moab points out that Windows Server 2008 added support for self-healing NTFS, which may detect some things. Still, that feature was new to Windows Server 2008, so it isn't something supported at all by Windows XP (or Windows Server 2003, or earlier). Even still, looking over the list of features, it appears that this involved some meta-data that helps the operating system notice problems that are so severe that the disk cannot mount, or other key areas of the disk that affect the operating system's kernel. It did not look like every single piece of data, in every single file, gets affected by this one particular feature.
The software for such operating systems is extremely unlikely to notice such things, unless they cause notable problems for the operating system to accomplish tasks. There may be some exceptions, like the portions of the operating system that check disks (CheckDsk/ChkDsk/ScanDisk/ScanDskW, depending on the operating system), but even they will be rather limited on what they can detect, largely because the filesystems don't store a very large amount of data that was intended to be useful for disk checking.
(RAID5 might be more prone to detect such things, with every bit having a parity bit that would help notice something unusual. Even then, it would be up to the RAID implementation to run a check to notice the problem. If the problem occurred on a part of the disk that isn't actively being worked with, the problem may remain unnoticed until someone tries to start using that data.)
In more recent times, larger numbers of bits meant that small likelihoods, like chances of "1 in 10 million", were more likely to affect things. The general public has also learned about "cosmic rays", which may have a small impact on things. Since bits are being crammed so tightly in newer devices, the physical requirements to represent a bit are smaller, so even small impacts are more likely to mess with how a bit is recognized. ReFS has some features designed to help with them being detected. Wikipedia's article on ReFS refers to this as "automatic integrity checking". As that is described as a notable feature of this filesystem, such features are likely more developed than with NTFS (and certainly more than FAT, which was comparatively simple in nature, and so had virtually no such features).