The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs $G$ and $H$, $G+H$ denotes the disjoint union of $G$ and $H$. This manuscript shows that (i) WIS can be solved for ($P_4+P_4$, Triangle)-free graphs in polynomial time, where a $P_4$ is an induced path of four vertices and a Triangle is a cycle of three vertices, and that in particular it turns out that (ii) for every ($P_4+P_4$, Triangle)-free graph $G$ there is a family ${\cal S}$ of subsets of $V(G)$ inducing (complete) bipartite subgraphs of $G$, which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of $G$ is contained in some member of ${\cal S}$. These results seem to be harmonic with respect to other polynomial results for WIS on certain [subclasses of] $S_{i,j,k}$-free graphs and to other structure results on [subclasses of] Triangle-free graphs.