We are given that
\begin{align*} \text{happy}&\Longrightarrow\text{can add}, &(1) \\ \text{purple}&\Longrightarrow\text{cannot subtract}, \hspace{15mm} &(2) \\ \text{cannot subtract}&\Longrightarrow\text{cannot add}. &(3) \end{align*}
Two solutions follow from here:

Solution 1.1 (Intuitive)
Combining $(2)$ and $(3)$ gives
\begin{align*} \text{happy}&\Longrightarrow\text{can add}, &(1) \\ \lefteqn{\underbrace{\phantom{\text{purple}\Longrightarrow\text{cannot subtract}}}_{(2)}}\text{purple}&\Longrightarrow\overbrace{\text{cannot subtract}\Longrightarrow\text{cannot add}}^{(3)}. \hspace{2.5mm} &(*) \end{align*}
Clearly, the answer is $\boxed{\textbf{(D)}}.$
~MRENTHUSIASM (credit given to abhinavg0627)

Solution 1.2 (Rigorous)
Recall that every conditional statement $\boldsymbol{p\Longrightarrow q}$ is always logically equivalent to its contrapositive $\boldsymbol{\lnot q\Longrightarrow\lnot p.}$
Combining $(1),(2)$ and $(3)$ gives \[\lefteqn{\underbrace{\phantom{\text{purple}\Longrightarrow\text{cannot subtract}}}_{(2)}}\text{purple}\Longrightarrow\overbrace{\text{cannot subtract}\Longrightarrow\lefteqn{\underbrace{\phantom{\text{cannot add}\Longrightarrow\text{not happy}}}_{\text{Contrapositive of }(1)}}\text{cannot add}}^{(3)}\Longrightarrow\text{not happy}. \hspace{15mm}(**)\] Applying the hypothetical syllogism to $(**),$ we conclude that \[\text{purple}\Longrightarrow\text{not happy},\] whose contrapositive is \[\text{happy}\Longrightarrow\text{not purple}.\] Therefore, the answer is $\boxed{\textbf{(D)}}.$
Remark
The conclusions in the other choices do not follow from $(**):$
$\textbf{(A) }\text{purple}\Longrightarrow\text{can add}$
$\textbf{(B) }\text{purple}\Longrightarrow\text{happy}$
$\textbf{(C) }\text{can add}\Longrightarrow\text{purple}$
$\textbf{(E) }\text{happy}\Longrightarrow\text{cannot subtract}$
~MRENTHUSIASM
Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that

all of his happy snakes can add,
none of his purple snakes can subtract, and
all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
$\textbf{(A) }$ Purple snakes can add.
$\textbf{(B) }$ Purple snakes are happy.
$\textbf{(C) }$ Snakes that can add are purple.
$\textbf{(D) }$ Happy snakes are not purple.
$\textbf{(E) }$ Happy snakes can't subtract.