For the most part the answer is no. Here are the exceptions:

If you are an elite member in either Celebrity or Royal Caribbean's program, the benefits carryover when you sail with the other line, but you will not receive credit in your "home" account for the cruise. Let me give you this example, you are a Platinum member in Royal Caribbean's Crown and Anchor program, and you decide to try Celebrity. When you get on board, your cruise card will show you as Select, and you will receive the benefits that a Celebrity Select member receives. You will then be credited with a Celebrity sailing, not a Royal Caribbean one. Note that you MUST have your travel arranger liase with the line to make sure that you get the proper benefits.

If you have sailed any of Carnival Corporations lines, you are considered a repeat passenger for all of their lines. So if you have sailed on Carnival, you can get a past passenger rate on Cunard.