Absolutely. With that quantity of sugar, you need a half cup (8 tablespoons) or more of butter to absorb all the sugar. The general ratio for creaming is 1/3 to 1/2 as much butter as sugar. So your recipe could reasonably have called for 1/2 cup to 3/4 cup of butter – but not the 3/16ths cup specified.
Now it may be that the recipe writer had some other tricks up his or her sleeve, reducing the fat content of the recipe by replacing some of the butter with a fruit purée, or some other alteration. But if so, he should have had you cream only a portion of the sugar with the butter, and mixed in the rest of the sugar later.
The point of creaming butter and sugar together is to whip as many tiny air bubbles into the butter as possible. These air bubbles provide the lift for the cake or cookies you are making. The leavening agent (baking powder or soda) inflates the bubbles as the batter cooks, causing it to rise. For this reason, some recipes call for you to cream them for as long as 5 to 10 minutes. If the butter and sugar do not come together, you won’t produce those tiny little bubbles, and your cookies or cake will be flat. Hence the importance of keeping the butter and sugar in balance.