The short answer is "No".
It can be higher in all cases.
It can be lower depending on the actual circumstances you have.
If the available short circuit current is less than the AIC rating of the 150A CB you can use it, even if the AIC rating is less than that of the 800 A CB.
If the 150A CB has a series rating with the 800A CB you might be able to take advantage of that.
Bob is correct on all counts, but I have to admit, the initial "No" got my hackles up for a moment!
The circumstances he mentions require investigation and information that may not be available to you or easy to get, so a simpler course would be to just match that rating of the main. But the issue is very valid, especially for mains, because often times larger frame breakers come with minimum AIC ratings that are higher than you need. So as long as the feeders below it are rated for the AFC (Available Fault Current), it's fine.
If there are other feeders in the panel, one way to quickly tell if the AFC is low enough or if there is a possibility of a series rating, is to look at the ratings of those other feeders. The risk however is that you are trusting the skill and judgement of the person who put n those in, and if they made a mistake, you are just continuing it. It's better to investigate it yourself.