Just to close this off - thanks to Mike-Doherty it appears that the answer is yes (and in fact for spaces this goes back to the paper "The decomposition of stable homotopy" by Joel Cohen.)

Using the terminology of Shipley's paper, given a sequence of maps $$ A_{n-1} \xrightarrow{f_{n-1}} A_{n-2} \xrightarrow{f_{n-2}} \cdots \xrightarrow{f_1} A_0 $$ then there is an object $X \in \{ f_1,\ldots,f_{n-1} \}$ if and only if $0 \in \langle f_1,\ldots,f_{n-1} \rangle$. The statement that $X \in \{ f_1,\ldots,f_{n-1} \}$ basically corresponds to the existence of the following commutative diagram:

$$ \begin{array}{cccccccccc} &&&& \Sigma^{-n+1}F_{n-1}X &&&& \Sigma^{-1} F_1X \\ &&&\nearrow & \downarrow &&&& \downarrow\\ \Sigma^{-n} X & \to & A_{n-1} & \xrightarrow{f_{n-1}} & A_2 & \xrightarrow{f_{n-2}} & \cdots & \xrightarrow{} & A_{1} \xrightarrow{f_1} & A_0 \end{array} $$

It is easy to see that the 3-fold bracket is just the usual Toda bracket. In addition the bracket constructed by Kochman is contained in the bracket constructed by Cohen.