Neurofibroma: A benign tumor that grows on nerve tissue, which can cause nerves to bulge out, especially in individuals with neurofibromatosis type 1. The description of thicker skin could be related to the tumor's growth pattern.

Varicose Veins: While not directly related to nerves, varicose veins can cause a bulging appearance under the skin, which might be mistaken for bulging nerves. The thicker skin description could be due to chronic venous insufficiency.
Lipoma: A benign tumor composed of fat tissue that can grow near nerves, potentially causing a bulge. The description could fit if the lipoma is close enough to the skin surface to cause noticeable thickening.
Peripheral Neuroma: A condition where nerve tissue grows abnormally, forming a tumor. This could cause a bulge and might be associated with skin thickening due to chronic irritation or inflammation.

Malignant Peripheral Nerve Sheath Tumor (MPNST): A rare but aggressive cancer that arises from the nerve sheath. It's crucial to distinguish MPNST from benign conditions due to its potential for metastasis and high mortality rate if not treated promptly.
Lymphoma or Metastatic Cancer: Although rare, certain types of lymphoma or metastatic cancers can cause nerve thickening or bulging. Missing these diagnoses could have severe consequences due to the potential for rapid progression.

Leiomyoma: A rare benign tumor of smooth muscle cells that can occur in the skin (cutaneous leiomyoma) and potentially cause a bulge. It's less likely but could be considered if other diagnoses are ruled out.
Neurilemmoma (Schwannoma): A rare, benign tumor that arises from the nerve sheath. While it can cause a bulge, it's less common than neurofibroma and might not typically cause the skin thickening described.
Plexiform Neurofibroma: A subtype of neurofibroma that can cause significant nerve and skin involvement, leading to a bulging appearance. This condition is associated with neurofibromatosis type 1 and can be disfiguring.