Yes, sometimes, kinda, not really is the answer. You must first understand that there are two main strains of Chinese writing currently in use, Traditional and Simplified Chinese. Traditional is used in Taiwan, Hong Kong, Macau and communities that are derivative thereof (usually the older Chinatowns and South Asian Chinese communities), while Simplified is use in Mainland China and later overseas communities that are more Mainland dominated. The Simplified system was formalized in the early Republic period, as part of the May 4th Movement in 1919, which was a nationalist response to western cultural and military invasions that involved reforming or shedding parts of Chinese culture considered weak, including the overcomplicated writing system. The system was made official in the 50s and 60s as part of the People's Republic of China's efforts to increase literacy rates, while the areas that weren't/aren't under the Communist Party's control retained the traditional characters. It is usually possible for a Traditional Chinese reader to read Simplified Chinese, as the characters are usually short hands already in use before its official implementations, or parts of existing characters that aren't too hard to figure out. The reverse is less true.

To go back a bit further in history, Chinese characters followed a slow evolution over some 4000 years to become what we know as "Traditional" Chinese. So the question can a modern Chinese person read "ancient" really depends on how ancient we're talking about, and who exactly is writing. Having done calligraphy as a child, a large part of which involved copying manuals written by famous calligraphists from the Tang, Song, Yuan and Ming dynasties, I can tell you that the standard fonts from the Ming period (Around 1300 AD onward) is near 100% readable for someone who can read Traditional Chinese, even at a schoolboy level, with readability deteriorating the further back you go. Some of the more cursive styles as written by cursive calligraphists are harder. Whereas characters from the Qin dynasty in 200BC are nearly illegible and would require study and quite a bit of guesswork. The Qin dynasty was considered the first to "unite" China (really just the heartlands of it) and was the first to unify the script, and therefore Chinese characters before that period would be even more impossible to read.

Note that all of this only concern the characters themselves. The reading of the characters and the formation of phrases and sentences change drastically over the years (worthy of a whole other post) and even between regions within China. So while a Chinese reader may understand the meaning of a character, it might not sound the same.

So, in conclusion: people of Chinese descent outside of China can read Chinese throughout most of history; people of Chinese descent inside of China can't read it as well unless they specifically study it; deciphering it gets harder and harder the older you get, until about 222 BC when it becomes nigh impossible because a unified Chinese script doesn't exist.