Hi,
1st thing to remember is that the 2960 & 3750 units may well not have enough memory to store two copies of the IOS file (in fact it's almost certain they don't). Therefor you'd have to delete the existing bin file before copying the new one.
Secondly, the bin file isn't normally stored in the root of flash on these units, instead there will be a subdirectory (usually of the IOS name) where it is stored, so the path would be something like flash:c3750-ipservicesk9-mz.122-55.SE10\c3750-ipservicesk9-mz.122-55.SE10.bin - you may want to look in that subdierctory to see what's actually in there.
To be honest, just copying the bin file isn't really the best way of upgrading the IOS these days; IOS now has the archive download-sw command which uses the equivalent *.tar file to upgrade the IOS as well as the web based management tools; it also takes care of changing the boot path for you. There's a config change template I wrote at https://thwack.solarwinds.com/docs/DOC-167187 which goes into more detail and also give you an idea of the commands etc.
Regards,
John