Could bilirubin increase be related with anxiety or depression?

No: Bilirubin levels r reflective of liver function.

Bilirubin: Depending on the level, could be indicative of biliary tract dysfunction (liver, gallbladder or bile ducts).

Not directly, BUT: Elevated bilirubin can point to liver, biliary, or pancreatic disease. Research shows a higher incidence of depression in liver disease (hepatitis b & c, cirrhosis, etc). http://www.ncbi.nlm.nih.gov/pubmed/21397104 also a link between pancreatic cancer and depression has been known for more than 70 years. Elevated bilirubin isn't a direct link to depression, but to other illnesses which may be.