In addition to what @Sweeper has mentioned: there is a natural tendency in Chinese to minimize the number of characters in compound words, e.g. 北京大学 becomes just 北大 in colloquial language. This is akin to abbreviations in other languages, e.g. California Institute of Technology vs. Caltech. But this point is only distantly related to your question.
To focus more on your exact question, you need to understand that 子 in 兔子 is just a "suffix" that you can use to build a myriad of things, like 勺子，筷子，儿子，包子 and so on. In itself 子 has a very abstract meaning and therefore you omit 子 in 兔子 when you build a compound like 兔年. The 兔 character already carries the proper meaning in itself alone, so there is no need for a redundant extra character. If I say that 子 is "redundant", you might as why on earth do we just use 兔子 instead of 兔. The reason is that modern Mandarin has a comparatively low number of possible syllables to work with, and therefore a huge number of homophone words, which make mutual understanding in oral communication quite hard sometimes. To solve this problem, modern Mandarin combines characters, in many cases of identical meaning, to build words that point to "simple" things like 街道 (street), 浸泡, 货物 are other examples of this. In other cases, it will use 子，儿 (erhua final 'r') or 头 like in 花儿 (flower) or 石头 ([a piece of] stone). In yet other cases a "prefix" like 老 as in 老师 or 老虎 will come in handy, or repeating the character, e.g. 哥哥.
But as soon as these two-syllable, "simple" words are combined with other ones, things start to get "economical", i.e. omit characters you don't need to express compound things. This is why we say 兔肉 instead of *兔子肉, 虎爪 (tiger claw) instead of *老虎爪, 大哥 (elder/st brother) instead of *大哥哥. The three character variants are just too long, the two character ones express the same meaning, and are therefore preferred.
You might now ask, if you can omit according to you own judging and sense. The answer is rather no. You really need to learn these things as vocabulary items, because you can shorten 火车站 to 车站 [which could refer to a railway station among others] but you cannot shorten 地铁站 to 铁站 or 地站. In this case all the two characters of 地铁 are need to express "subway", so omitting characters does not apply here. On the other hand 港口 (harbor, port) becomes just 港 in compounds like 港水 (harbor water) or 海港 (sea port) and not 港口水 or 海港口.
UPDATE: OK, 铁站 or 地站 might work in some cases, especially when they are part of another compound expression, but 地铁站 sounds more standard.
