"I think my translation is right."
Here 的 is not a possessive article, it is a attributive marker. Actually, you can consider the "possessive particle" usage as a special case of attributive marker. For example, "我" = I (nominal pronoun), "我的" = my (adjective pronoun). That is the usage of attributive marker - making the word / phrase before it an adjective.
We can look at the examples below:
红（色）是一种颜色。 Red (color) is a kind of color.
红（的）汽车 red car
这辆汽车是红的。 This car is red.
You can see that the word "red" is used as a noun in the first sentence and an adjective in the second and third sentence. If you look the word "red" in the dictionary, you may see that the word can be both a noun and an adjective. Does it really reflect the internal logic of the language? I'm not sure. But it still works.
But in Chinese, 红 itself is not a real adjective (and actually not a real noun too). You need a mark to make it an adjective. You can guess what this sentence means:
This very awkward sentence means "This car is red" too. But the "red" here is the noun "red", the concept "red". That's why "我觉得我的翻译是对" is wrong.
Other attributives, including those who don't look like adjectives at all (like the complicated example below), need the attributive marker 的 too:
a method (that simulates light propagation in media (whose optical properties are variable in space))
(在 (光学性质在空间中可变的) 介质中模拟光传播的) 一种方法
See, those lengthy clauses are treated as adjectives too. If you replace the content in the bracket with any other adjective, it is still a valid sentence.
In an adj. + n. phrase, the marker 的 can be omitted (as in the second example). The rules are complicated but still similar to English. You can find a detailed description here: https://www.duolingo.com/comment/25480888
That is a great explanation, Warren! Robert, it may help to look at it this way: the final noun is omitted becsuse it is understood. You can say "我觉得我的翻译是对的翻译" meaning "I think my translation is the/a right translation." In that sentence, you can see that 的 is needed in 对的翻译 to make 对 an attributive/adjective describing 翻译. However, we don't really need to say "translation" twice, in English or Chinese. That sentence is a bit wordy. You can leave out the second "translation" and it's still understood that that is what we're talking about. In English that becomes "I think my translation is right." In Chinese that becomes 我觉得我的翻译是对的。 对的翻译 becomes 对的 because 翻译 is understood.