Dependability is a key factor in any software system due to the potential costs in both time and money a failure may cause. Given the complexity of Grid applications that rely on dependable Grid middleware, tools for the assessment of Grid middleware are highly desirable. Our past research, based around our Fault Injection Technology (FIT) framework and its implementation, WS-FIT, has demonstrated that Network Level Fault Injection can be a valuable tool in assessing the dependability of traditional Web Services. Here we apply our FIT framework to Globus Grid middleware using Grid-FIT, our new implementation of the FIT framework, to obtain middleware dependability assessment data. We conclude by demonstrating that Grid-FIT can be applied to Globus Grid systems to assess dependability as part of a fault removal mechanism and thus allow middleware dependability to be increased.