Reliability analysis of flow meters is an important issue for process industry companies because of the need to ensure the production quality and operational safety. In practice, field data for flow meters failure process can have complicated structure due to multiple failure modes arising from different electromechanical parts. Besides, incomplete records generally exist. For example, the installation date is usually not available, making the failure data left-truncated. There also exist right-censored cases since many units are still in service when the data are analyzed. In this paper, we use a nonhomogeneous Poisson process model with power-law intensity functions to address multiple failure modes for field data with both left-truncated and right-censored cases. We apply the maximum likelihood method to estimate the model parameters. In order to address the statistical uncertainty, random weighted likelihood bootstrap procedure is used to estimate the standard errors and confidence intervals of the parameters. Real-world flow meters failure data from a process industry company are used in the case study. Estimated intensity functions and estimation of mean time to failure are obtained and show that the parametric model can reasonably fit the failure data well.