Breaking in from a non-target school is absolutely possible, but it usually requires a more intentional process. You cannot rely on the school brand to open doors, so other parts of the profile and process need to carry more weight.
The biggest levers are typically a strong resume, thoughtful networking, and excellent interview performance. Networking matters because it increases your information flow and can help your application get better context, but it does not replace the need for a genuinely competitive profile.
You should think of it less as an impossible path and more as a path with less margin for error. The stronger and more coherent the overall story, the less the school label dominates the conversation. You may also find it helpful to read breaking into consulting from a non-target school if you want a more structured way to practice this.