Sequencing problems with an unknown covering or packing constraint appear in various applications, e.g., in real-time computing environments with uncertain run-time availability. A sequence is called α-robust when, for any possible constraint, the maximal or minimal prefix of the sequence that satisfies the constraint is at most a factor α from an optimal packing or covering. It is known that the covering problem always admits a 4-robust solution, and there are instances for which this factor is tight. For the packing variant no such constant robustness factor over all instances is possible. However, in both cases, many problem instances allow for a much better robustness guarantee than the pathological worst-case instances. In this work we aim for more meaningful, instance-sensitive robustness guarantees. We present an algorithm that constructs for each instance a solution with a robustness factor arbitrarily close to optimal. We hope that the idea of instance-sensitive performance guarantees inspires to revisit other optimization problems and design algorithm tailored to perform well for each individual instance.