Fence bottom rotting between boards

i?m curious about my fence construction and why we are getting rot at bottom. i have a solid cedar fence with the vertical 1x6 pickets set 1/2 inch apart. these are held in place by a horizontal 2x4 at top and bottom. there is also the addition of a 1x 3 trim cover board on the neighbor side of fence at bottom. and this is where i have a problem. the pickets are rotting where they are sandwiched between these 2 boards. the bottom of fence is 5? off ground so it is not soil contact. my questions are one. Is this normal construction for the bottom of a cedar fence? seems to be a weak spot because debris and water can soak in there. how can i repair this without replacing all the vertical boards? i was thinking i could attach a new 2x4 above the old one and then cut off the rotted pickets.