I saw a etching of The Temple of Vespasian where the columns of the temple are buried almost to the top. If I recall in this specific instance: the Roman Forum is in swamp land, and the Tiber had flooded bringing silt into the forum. However, the Parthenon and Largo Di Torre Argentina are ~4-5 m below 'modern' street level, and are not on swamp land.
How do sites get buried generally? How are some sites unburied after thousands of years (Stonehenge) and others completely buried after a couple hundred (Tenochititlan, covered in this case)? I can understand unused roads becoming buried by a couple of cm but not whole meters.
This depends on geologic and climactic forces. Some parts of the world collect sediment at certain times while others do not.
Regarding Rome, the city lies on the Tiber River which deposits silt, slowly burying the city over time.
In other cases, the land is rising or falling due to deeper mechanisms. Much of the coastline existent 20,000 years ago has been sunk beneath rising sea levels; while at the same time landmass that once was pressed down by glaciers begin to rise.
Other parts of the world sink or expose artifacts due to plate tectonics. Two opposing examples: the American south west and Venice, Italy. Venice is sinking due to rising sea levels but also because the alps are pressing the lagoon into the earth. The southwest however is a rising mountain plateau. Dust is blown off these high areas exploring artifacts both human and plant or animal.
I've done archaeolgical surveys where I found a Pepsi can three feet deep. I have been on an archaeological dig where we found 2000 year old archaeolgical features 1.5 feet deep. The former was on river floodplain where a lot of sediment and organic mateiral gets deposited every time the river floods. The latter was on a hilltop overlooking a small river valley and soil would only be deposited via wind or agriculture at a very slow rate. That's just a couple examples of why materials can be found at very different depths independent of age.
Consider /r/AskScience as well.