Consider the expression ab+bc+cd. If a and d were zero then b+c=1, so let b=c=1/2. Then the expression=1/4. Note that b and c appear twice: b in ab and bc, and c in bc and cd, while a and d appear only once each. Therefore b and c contribute more to the sum than a and d. Although none of the quantities can be zero, a and d can be sufficiently close to zero so that the expression approaches 1/4. If b=1/2-x and c=1/2+x then bc=1/4-x^2 so the product is always <1/4. If b=1/2-x and a=x, and c=1/2-x and d=x, where x is very small and positive, a+b+c+d=1. The expression becomes: 2x(1/2-x)+1/4-x+x^2=x-2x^2+1/4-x+x^2=1/4-x^2. Therefore the maximum value is <1/4, but x can be made infinitesimally small. This makes the maximum 0.25.
Example: x=1/100: a=d=0.01; b=c=0.49; expression=0.0049+0.2401+0.0049=0.2499.
Example: x=1/1000: expression=0.249001+2*0.000499=0.249001+0.000998=0.249999.