In structural health monitoring (SHM) systems for civil structures, massive amounts of data are often generated that need data compression techniques to reduce the cost of signal transfer and storage, meanwhile offering a simple sensing system. Compressive sensing (CS) is a novel data acquisition method whereby the compression is done in a sensor simultaneously with the sampling. If the original sensed signal is sufficiently sparse in terms of some orthogonal basis (e.g., a sufficient number of wavelet coefficients are zero or negligibly small), the decompression can be done essentially perfectly up to some critical compression ratio; otherwise there is a trade-off between the reconstruction error and how much compression occurs. In this article, a Bayesian compressive sensing (BCS) method is investigated that uses sparse Bayesian learning to reconstruct signals from a compressive sensor. By explicitly quantifying the uncertainty in the reconstructed signal from compressed data, the BCS technique exhibits an obvious benefit over existing regularized norm-minimization CS methods that provide a single signal estimate. However, current BCS algorithms suffer from a robustness problem: sometimes the reconstruction errors are very large when the number of measurements K are a lot less than the number of signal degrees of freedom N that are needed to capture the signal accurately in a directly sampled form. In this article, we present improvements to the BCS reconstruction method to enhance its robustness so that even higher compression ratios can be used and we examine the trade-off between efficiently compressing data and accurately decompressing it. Synthetic data and actual acceleration data collected from a bridge SHM system are used as examples. Compared with the state-of-the-art BCS reconstruction algorithms, the improved BCS algorithm demonstrates superior performance. With the same acceptable error rate based on a specified threshold of reconstruction error, the proposed BCS algorithm works with relatively large compression ratios and it can achieve perfect lossless compression performance with quite high compression ratios. Furthermore, the error bars for the signal reconstruction are also quantified effectively.