Are they of a tongue-and-groove design? Or are they just supposed to drop into a support frame?

Since you emailed me that it was a support frame, my answer is that normally they just drop in. I'm wondering if the new tiles are just a little bit different in dimensions from what you have. Generally these tiles are interchangeable, but, if the factory made theirs 1/16" oversize compared to what you have, and your support frame is a tight fit to start with, it could be a problem, especially if you try to insert several tiles adjacent to one another. Do a careful measurement of the part of the tiles that drops through the frame and compare the dimension to what you have. If the new ones are larger, your choices would be to get a different brand, or to run the tile across a table saw with a slightly raised blade to slightly reduce the dimensions of the oversized area that is supposed to drop through the frame. (That would mean having one or two exposed edges, though, which you would have to try to hide with white paint.)

Sorry, that's the best I can suggest.