Cost over runs can happen at all points in the life cycle - incomplete requirements, poor design, lack of skills to manufacture the design, inadequate testing, etc. All these are risks which can be mitigated - have a requirements review the designers participate in, have an expert review the design, allocate suffiicient project time for testing, provide training, etc. and be sure these are built into the budget. Also make sure there is adequate contingency budget and time.
In general, the project manager is accountable, however the project team is responsible for working together with the project manager to identify the risks and mitigation strategies and doing their part to avoid them. If there is a requirements review, the project team must be prepared to diligently review the requirements. In a sense, these are all shared responsibilities which happen at a point of hand off. If the requirements are unclear, the designer needs to ask questions and validate their understanding such that they can design to the requirements. The project manager needs simple facilitation skills to bring together the team in planning to address these.
At the end of the day, hopefully management is looking for the root cause (if there is a lack of skill, providing training, if the requirements aren't being diligently reviewed consider if other resources are required) and everyone is pulling together for project success.