A function f∈L^2(R) is said to have bandwidth Ω>0, if Ω is the maximal frequency contributing to f. The concept of variable bandwidth arises naturally and it is even more intuitive when we think about music. Indeed, the perceived highest frequency, i.e. the note, is obviously time-varying. This observation provides a reasonable argument for the assignment of different local bandwidths to different segments of a signal when representing it mathematically. However, producing a rigorous definition of variable bandwidth is a challenging task, since bandwidth is global by definition and the assignment of a local bandwidth meets an obstruction in the uncertainty principle. We present a new approach to the study of spaces of variable bandwidth based on time-frequency methods. Our idea is to start with a discrete time-frequency representation that allows us to represent any f as a series expansion of time-frequency atoms with a clear localization both in time and frequency. We may then prescribe a time-varying frequency truncation and, in this way, end up with a space of a given variable bandwidth. For these spaces, we study under which sufficient conditions on a set of points a function can be reconstructed completely from the evaluation of the function at these points. Analyzing some MATLAB experiments, we motivate why these new spaces could be useful for the reconstruction of particular classes of functions.