Is it possible to have a positive blood test, but not have celiac disease?
Yes, blood work can be falsely positive, as can any test in medicine, especially at low titers. However, it may be that the biopsy was done or read incorrectly or that there’s not yet enough damage in the small intestine to diagnose celiac disease. A second opinion at a celiac center can help clarify a diagnosis.
False positives vary between the tests and also between the populations considered. For instance, tTG is falsely positive in about 2-3% of people in general, but in about 20% of those with Type 1 Diabetes or other autoimmune conditions. EMA, on the other hand, has basically no false positives.
You could consider a genetic and EMA test to assist with your diagnosis. In certain cases these can provide a strong enough indication to diagnose the disease when in conjunction with a positive response to a gluten-free diet.