Said a simpler way:

1/3= 0.333…

1/3 + 1/3 = 0.666… = 0.333… + 0.333…

1/3 + 1/3 + 1/3 = 1 = 0.333… + 0.333… + 0.333…

The quirk you mention about infinite decimals not incrementing properly can be seen by adding whole number fractions together.