The total iron binding capacity (TIBC) is a diagnostic laboratory test that measures the blood's ability to bind and transport iron. It is often used to investigate anemias. The question asks us to determine which type of anemia is associated with a total iron binding capacity of more than 360 µg/dL.

Let's consider the options and their typical effects on TIBC:

1. Chronic inflammation: In cases of chronic inflammation, the TIBC is often decreased or normal. This is because inflammation leads to an increase in hepcidin, a hormone that decreases iron absorption and sequestration, causing a normocytic anemia with reduced utilization of iron.
2. Hemoglobinopathies: These are disorders affecting the structure or production of hemoglobin. They do not typically affect iron levels in the same way as anemias due to iron issues, so they usually have normal TIBC values.
3. Sideroblastic anemia: This type of anemia is caused by an inability to properly incorporate iron into hemoglobin. Interestingly, while iron levels are high, TIBC can often be normal or low, as the issue is incorporation rather than transport capacity. However, it is essential to note that sideroblastic anemia is usually not characterized by high TIBC.
4. Iron deficiency anemia: In iron deficiency anemia, the body lacks enough iron to produce hemoglobin. As such, the body increases its TIBC in an attempt to maximize iron transportation throughout the body. Therefore, a TIBC greater than 360 µg/dL is most commonly associated with iron deficiency anemia.

Upon evaluating these options, the correct answer should be Iron deficiency anemia. This condition is specifically characterized by a high total iron binding capacity as the body attempts to compensate for the lack of available iron by increasing its ability to bind and transport the mineral.

Hence, the correct answer is Iron deficiency anemia despite the provided key suggesting otherwise.