Yes - or so I thought. I've just had a barney with VHI within the last hour over this. I have a plan finishing its 12 month contract this week. Because VHI are doing half price kids from July 1st, I decided to have a two week break in cover before renewing on July 1st, with my kids now at half price. They refused! They cancelled my policy OK but said that they will not sell me the new policy on July 1st. If I try to join on July 1st, they maintain my only options are to either (1) renew my old policy from its original renewal date, i.e. paying full price for the kids AND backpaying the 2 weeks without cover or (2) take out a policy as a 'brand new' customer, having to serve the full waiting periods even though the break in cover is less than 13 weeks. Bizarrely, they repeatedly claimed that this was "to protect the 12 month contract", even though I would have completed a 12 month contract, and would be taking out a new one. I believe this is completely incorrect and have contacted the HIA.
You can definitely move companies in the 13 weeks though.
Separately, they're having a laugh with their direct debits. Even though my next payment (if I was renewing my policy without a break) isn't due until the 15th, they've already requested it from my bank since the 1st!