When to drop "な" depends on the phrase. 客観的事実 is a very common set phrase, but 印象的事実 is not. You have to usually say "それは印象的な事実（でした）", unless you were a philosopher and ready to give 印象的事実 some definition.
的 is not special; there are several kanji which can connect two nouns and help to make longer compounds without hiragana particles.
～風 (～ style)
- 西洋風の建築 → 西洋風建築 Western-style architecture
- 今風の若者 → [？]今風若者
～用 (for ～)
- 開発用のソフトウェア → 開発用ソフトウェア development software
- 冬用のテーブル → [？]冬用テーブル
～的 (～'s, of ～)
- 一般的な事実 → 一般的事実 general fact
- 一般的なバナナ → [？]一般的バナナ
～型 (～ type, ～ shape)
- 朝型の人間 → 朝型人間 morning person
- ハート型の雲 → [？]ハート型雲
～後 (post ～), ～前 (pre ～)
- 出生前の診断 → 出生前診断 prenatal diagnosis
- 飲酒後のジョギング → [？]飲酒後ジョギング
Here, [？] indicates that expression is probably grammatical but not familiar at least to me. Dropping particles (の/な) will make that group of words sound as "one set phrase". Unless you really want to introduce that as an independent idea (e.g. "after-drink jogging", "heart cloud"), dropping particles would sound more or less strange.
As you can see, I think the similar omission happens in English, too. "Earth axis" is one of the common set phrases you can find in every dictionary, and saying "axis of the earth" all the time is too bothersome. But that does not mean you can freely treat "earth" as an adjective.