In competing risks data, in practice, there may be lack of information or uncertainty about the true failure type, termed as 'missing failure type', for some subjects. We consider a general pattern of missing failure type in which we observe, if not the true failure type, a set of possible failure types containing the true one. In this work, we focus on both parametric and non-parametric estimation based on current status data with two competing risks and the above-mentioned missing failure type. Here, the missing probabilities are assumed to be time-dependent, that is, dependent on both failure and monitoring time points, in addition to being dependent on the true failure type. This makes the missing mechanism non-ignorable. We carry out maximum likelihood estimation and obtain the asymptotic properties of the estimators. Simulation studies are conducted to investigate the finite sample properties of the estimators. Finally, the methods are illustrated through a data set on hearing loss.