Of course you can use `montage` from the _Imagemagick_ suite of utilities. The image has a 3:4 aspect ratio; preserve that in the geometry setting to avoid cropping related issues and borders. It's just about repeating that file name as input in the command, and basic arithmetic. Maybe there was something in the manual, but otherwise, assuming the 4x4 "tiles" example, you can try:
montage $(printf 'yourfile.tiff%.0s ' {1..16}) -geometry 192x256 -tile 4x4 -unsharp 0x.5 output.tiff
The command substitution is unquoted to generate the required tokens. With a final touch of _sharpen_.