This work revisits the formulation of interpolation sequences, in order to better understand their relationships with Bounded Model Checking and with other Unbounded Model Checking approaches relying on standard interpolation. We first focus on different Bounded Model Checking schemes (bound, exact and exact-assume), pointing out their impact on the interpolation-based strategy. Then, we compare the abstraction ability of interpolation sequences with standard interpolation, highlighting their convergence at potentially different sequential depths. We finally propose a tight integration of interpolation sequences with an abstraction-refinement strategy. Our contributions are first presented from a theoretical standpoint, then supported by experimental results (on academic and industrial benchmarks) adopting a state-of-the-art academic tool.