Ken, many times I divide my large Cattleyas and sometimes there is no problem and sometimes I outright kill my plant.
I have to look back a few years, I had a huge Copper Queen in a wooden basket well established and it was getting downright unmanageable. The roots were hanging down about 3 feet long, no joke. And there was the problem, the way the plant was anchored, I just couldn't make proper cuts without major damage. To make a long story short, all three divisions slowly but surely perished on me.
Fire Dance 'Patricia' can be split withought trouble, probably it grows in all directions, also out of the middle of the clump.
Cattleya skinneri likes to grow into a ball of sorts, now when you cut it, you can't avoid taking along healthy eyes and growth. It took years for my divisions to really pick up steam again.
I am always very careful nowadays how and at what stage I divide. To me it isn't that straight forwards. I am always in awe how professional growers like Silva split off divisions without batting an eyelash.
This Spring I cleaned up a lot of plants and repotted many. So far so good.