Is it possible to get a measurement model with high reliability and validity while your structural model fit is poor?
Yes, it could possible because the reliability and validity are related to indicator loading. Once you have high indicator loading, you can get high reliability and validity values as well.
There are three validity measures of a construct namely, Construct Validity, Convergent Validity, and Discriminant Validity.
Construct Validity is measured using fitness indexes but Pls-SEM do not have such indexes. Convergent Validity is measured using AVE and computed using factor loading. Same for Construct Reliability also measured using Composite Reliability (CR) and AVE where like AVE, the CR is also computed using factor loading.
The measurement model might be valid, but not Construct Validity because there is no fitness indexes to gauge that. So the answer might be Yes and No.
Yes. Unless the structural model is saturated, it can have poor fit. Poor fit indicates that it is inconsistent with the data. Perhaps simply because it's incorrect (note that a saturated model can also be incorrect). It's a separate question from measurement.
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