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That's what I wanted to say at first, but after analysing some imaginary situations I have to admit that sometimes "Тише!" pops up in mind instead of "Осторожнее!", probably beacause it's way shorter...

Yes. it is true. It is more related to soviet times. For security reason ( or they thought that it can improve security..) mail addresses of such enterprises
read as 'п/я ###' ( почтовый ящик )...

Тише!/Тихо! - yes, especially when the person stumble over or drop something, or in any situation, where the quick reflexes are needed. Sometimes Осторожнее!/Осторожно! is more appropriate...