Whether a good is a Giffen good should be based on circumstance? The textbook example of a Giffen good is the potato during the Irish Potato Famine. It is characterised by a positive income effect that is larger than the negative substitution effect when the price of the good increases. During the famine, the increased price of potatoes counterintuitively led to an increase in the quantity consumed of potatoes. This was because potatoes were likely one of the only available staples for the peasants and that alternatives may have been significantly more expensive. Hence, the increased consumption of potatoes and a diversion away from more expensive foods was observed during the famine. This makes sense in the context of the peasants having very low, limited incomes and that other staples or foods were either unavailable or far more expensive. 

If the circumstances were changed and say, rice or noodles, were cheaper than potatoes, I am certain that there would be a drop in the quantity demanded for the potatoes given the increase in price. Hence, I would like to ask if it is possible to assert that a good can be considered a Giffen good only under the appropriate circumstances? That is, the income and substitution effects would play out differently given different circumstances and under certain circumstances, these effects would be opposite with the income effect being larger. 

In fact, if you think about it, given a sufficiently large increase in the price of potatoes, the quantity demanded would have to decrease as it really becomes unaffordable and the peasants may have to resort to consuming less of it and find some other substitute. Hence, the Giffen good behaviour would only be observed for a particular price range and not below or above that.
Yes, whether a good is a Giffen or inferior good depends on "the circumstances", or to be more narrow, it can depend on the income and price levels.

Claim 1. Being an inferior good is a local attribute.

Proof. Suppose a good $x$ is inferior for all income levels $I$ and all price vectors $\mathbf{p}$. Trivially demand for $x$ at $I = 0$ is 0. If $x$ is inferior for all income levels, then for any $I &gt; 0$ demand for $x$ is also 0. This contradicts the strict definition of inferior good, which specifies that as $I$ increases $x$ should decrease.

Claim 2. All Giffen goods are inferior goods.

This is covered in some detail in textbooks, usually in a chapter that mentions the "Slutsky-equation".

Combining the two claims we have that being a Giffen good is also a local attribute; as being Giffen everywhere would imply being inferior everywhere, and we have shown that that is not possible.