We did a 10 night trip at the end of Jan/start of Feb. My middle child was coming down with something on the last day. (We found out she had a double ear infection the next day). My kids were 11 months, 3, and 6, so we actually beat the odds by having no one sick for that long of a trip. The next year, while at home, we had 6 weeks straight of someone being too sick to leave the house between the three of them. So, you never know! That being said, we've done many Disney/Universal trips and only had illness get in the way of our plans once. (We didn't know about the ear infection on the other trip - just that our 3 year old wasn't acting like herself, but we didn't know if that was just being tired at that point). My eldest came down with a stomach virus the day before we were scheduled to leave. Again, this was the end of a 10 night trip. Our planned park was one we had already done it so figured we would just all head home a day early. It wasn't a big deal.
On a plus side too - the end of Jan/start of Feb was the lightest crowds I've ever seen and we tend to go during off-seasons as it is.
With 6 kids, the chance of someone getting sick is of course greater. But that doesn't mean it will happen - or that you can't work around it. Many of your kids are older, making it easier to split up if you need to.