I got a translation of “Are hospitals in your country good?” from a native speaker of Mandarin as:
Why there is no 的 after 你? It is after a pronoun for instance in such a sentence:
This question is really about: " when can we omit the possessive 的 "
的 1.(adjective suffix):
强大勇敢 (strong and brave)
强大勇敢的人 (strong and brave man)
You cannot omit the adjective suffix 的 and write 强大勇敢人
高大 (tall and big)
高大的人 (tall and big man)
You cannot omit the adjective suffix 的 and write 高大人
Adjectives that do not need adjective suffix can directly apply to the object. For example 好(good) in 好女儿(good daughter)
我(的)衣服的鈕扣的顏色是白色的。 can be reduced to: 我衣服鈕扣的顏色是白色的。
(The last 的 is not omitted because it is not a possessive 的 but an adjective suffix 的.)
The same with your example sentences:
你(的)国家的医院好吗？can be reduced to: 你国家的医院好吗?
我(的)妈妈的房子很高。can be reduced to: 我妈妈的房子很高。
From user3a's answer:
The “的” before the subject is usually kept.
Others can be omitted, if possible.
I say it is wrong. The possessive '的' after a subject is usually kept, if it is a 'true possessive'(something you actually own) For example: The possessive 的 in 我的車 cannot be omitted and make it "我車"
On the other hand, if the '的' is a 'relative possessive'(something you don't actually own) can be omitted. For example: you can omit the '的' in '我的母亲' and write 我母亲.
The possessive 的 in 这是我的太太; 他是我的女儿; 你的国家; 你的公司 can be omitted. You can write 这是我太太; 他是我女儿; 你国家; 你公司 (if you actually own the country or the company, then the possessive 的 should be kept)
Again with your example sentence:
我(的)妈妈的房子很高。 can be reduced to: 我妈妈的房子很高。
我(的)牛(的)角很长。 cannot be reduced to: 我牛的角很长。 or 我的牛角很长。
(you possess the cow and the cow possesses the horns, neither can be omitted.)
I believe that 的 is omitted because it's not in the front of the subjects or objectives and because it's not fluent to use too many "的" in a sentence.
In TangHo's answer the "strong 强大的" and "tall 高的" are also adjectives.
Why did he omit the "的"?
The "的" in "好(的)女儿" is omitted because it's not fluent either.
I don't think it's due to "do not need adjective suffix".
Although "好女儿会很孝顺" is more fluent and simple, I think "好的女儿会很孝顺" is also acceptable.
In the user3a's answer he/she said "The“的” before the subject is usually kept."
But TangHo quoted as "after".
I don't think it's a mistake because we usually do the "copy and paste" when we quote something.
As I know, the "的" in "这是我的车" can also be omitted.
I think "这是我车" and "我牛的角很长" are acceptable in the verbal talking.
You can ask your Chinese friends, MrVocabulary.
(a) does sound "un-Chinese" For example when answering a question, like, 这是谁的车? you answer 车是我的, and not 是我车.
Coming back to the question, why can't we say 你的国家的医院好吗?
I think if you really want the 1st 的 there, the sentence might end up as:-