If I recall correctly, the stainless reed is no thicker than the original phosphor-bronze ones. However it is really easy to not separate a couple or even three of them and install them as one. Use a solvent and magnifier to be sure you only have one reed to install! The oil bonding the layers together is pretty thick and congeals with time, and the reeds are only .0015" thick, so several look like one.
Having worked on the development of them, they are the best we could devise. And be sure we had all the historical ones to check out! I personally have never been able to make the teflon reeds work well at all, and the Mylar ones are superb at first, but warp with heat and have no durability.