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Why do I get convergence error when I increase the mesh?

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I solved a model with low mesh like 11. The results were not satisfactory at all but no error. Then I increased the mesh to 6000, and it stuck in 4% and after 20 minutes I got the convergence error. However, when I chose a 160 mesh it took me four days to solve, and the results are satisfactory for the same model. I did not change anything here except mesh. Why the increased number of mesh to 6000 gave convergence error? It does not make sense to me. It should be opposite of that.

Here is some more information:
Model is Time dependent (0, 0.1,7)
Four different physics are running in my model.
I used Edged mesh with distribution properties of "fixed number of elements."
I sometimes get this warning too "Minimum element quality: 1.031e-011 Warning: Low minimum element quality."
or this warning "Warning: The number of allocated threads (4) exceeds the number of available physical cores (2)"
The reciprocal of size varied from 10^5 to 900 or maybe near zero.
The complete mesh consists of 2754 domain elements and 384 boundary elements.

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