Violation of completeness axiom (simple everyday examples)
There is a problem in how you translate completeness into behavior. Let $R$ be any binary relation, representing preferences, on a set $X$ of alternatives and $A\subseteq X$ be a nonempty set of alternatives available. The usual assumption is that the decisionmaker chooses an alternative $a\in A$ optimally according to the relation $R$. Here are three ways to interpret this:


$a R b$ for all $b\in A$.
$b R a$ for no $b\in A$. 
$b R a$ implies $a Rb$ for all $b\in A$.


One usually uses 1. for weak (reflexive) preference relations, 2. for strict (irreflexive) preference relations, and 3. works in both cases. If we define $R'$ by $xR'y$ iff $\neg(yRx)\vee(xRy)$, then $R'$ is complete and satisfies 1. iff $R$ satisfies 3. If $R$ is irreflexive, $R'$ satisfies 1. iff $R$ satisfies 2. So completeness in itself has no behavioral implications from this perspective. It can interact with other properties. For example, transitive and complete is more restrictive than transitivity alone- completeness forces transitive to be applicable to more comparisons.

Now one can try to make a distinction between indifference and incomparability. But such a distinction necessarily uses a different framework than the usual maximization paradigm.