Spectrometers are widely used for characterizing materials. Recently, filter-based spectrometers have been pro- posed to lower the manufacturing cost by replacing optical components with low-cost wavelength-selective filters, but at the expense of possibly lowered signal quality. We present compressive spectrometers which, based on the compressive sensing principle, are able to recover signal with improved quality from measurements acquired by a relatively small number of low-cost filters. We achieve high quality recovery by leveraging the fact that spectrometer measurements typically follow the shape of a smooth curve with a few spikes. We validate our method with real-world measurements, and release our dataset to facilitate future research.