<p>Uncertainties complicating the real-world problems seriously impact the decision-making process and create a demand for customized tools to address specific concerns. While the possibility fuzzy soft sets effectively handle decision-making problems, by incorporating both the unknown prevalences of particular impreciseness as well as their impact, the problems like bipolar disorder diagnosis, characterized by contrasting criterion, seek additional handling for opposing parameters. Such problems are addressed in this work by proposing the possibility fuzzy bipolar soft sets (POFBSSs) as a merger of bipolar soft sets and possibility fuzzy soft sets. The resulting hybrid model combines the decision-making power of possibility fuzzy soft sets with the bipolar parameter handling feature of bipolar soft sets. Corresponding to the proposed model, the set-theoretical notions like subset, absolute and null sets, complement, union, and intersection are discussed. Novel multiple criteria decision-making (MCDM) algorithms based on the corresponding OR and AND operations are proposed and implemented on an investment problem for maximizing profit potential and minimizing risks. Similarity measures based on POFBSSs are defined and numerically illustrated. A detailed diagnosis of bipolar and related disorders is performed in a model scenario using POFBS similarity analysis which addresses the gaps motivating this work. Finally, advantages and disadvantages discuss the improved uncertainty depiction against the existing models as well as the room for further improvements.</p>