I'll assume that you mean whether this was known by the time Europeans arrived, because otherwise - to stick with the 20-year rule - to ask whether Native Americans were aware of the geography of the Americas in 1996 is trivial.

Not only is when important, where is also important. It's much easier for the Kuna of Panama to know about both coasts than it is for the Lenape in New Jersey. That said, the Lenape - like many other nations in the Eastern Woodlands - conceived of North America as a vast island named for the primordial turtle that carries it on its back. Modern ethnogeography of the "Turtle Island" persuasion goes further, associating specific parts of North America to various parts of the primordial turtle (the Rockies and the Appalachians are the ridges on its shell, Central America is its tail, etc.) But is this all just a coincidental lucky guess?

In the late 1800s, Daniel Brinton was working with Lenape living at the Six Nations of Grand River in Ontario to compose a Lenape dictionary, during which time he received this bit of information:

The old men still relate with pride that, in the good old times, before any white man had landed on their shores, "the Lenape had a string of white wampum beads, wapakeekq', which stretched from the Atlantic to the Pacific, and on this white road their envoys traveled from one great ocean to the other, safe from attack.

Here, the Lenape metaphorically entangling white wampum - a sign of peaceful intention - with roads and the right of passage through foreign territory. From the Mississippi River into the Plains, the calumet (peace pipe) could be used to serve the same function. But Brinton got this information in the late 19th Century when this sort of geographic knowledge was common place. What evidence is there to support that the Lenape could have been crossing the the continent prior to the arrival of Europeans?

The Hopewell period (about 200 BCE to 400 CE) was an era of particularly far reaching trade routes across what's now the US. Among the artifacts demonstrating long-distance trade, obsidian has been found in New Jersey that has its origins in California. How it got to New Jersey isn't fully understood - did an individual bring it all the way across the continent or did it get passed from one group to another until it couldn't go any further east? An individual journey may be more likely. We know that other people in this era were almost certainly make long distance journeys to acquire obsidian (the Scioto Hopewell in Ohio made periodic trips to Yellowstone for their obsidian, as well as picking up some rare flints in North Dakota along the way), and we don't see the typical fall-off patterns associated with down-the-line trade linking California and New Jersey.

From the early colonial period, we do have a few stories about individuals making such a journey. The most famous and detailed is that of Monacht-ape, which I discussed here. Late in his life, sometime during the 1720s, he was interviewed by the French historian du Pratz - so his journey probably took place in the later decades 1600s. He first traveled from his home in northern modern-day Louisiana to New England and back again. A little later, he undertook a second journey to the west, traveling up the Missouri River and back down the Columbia until he reached the Pacific coast. He traveled north along the coast for a while, until he eventually heard about the Bering Strait and decided to return home.

This isn't the only reference du Pratz to people in the lower Mississippi knowing about the Pacific. He also interviewed a Natchez priest who was charged with keeping the nation's history. A notable event in this history involves the war between the "Warriors of Fire" (ie the Spanish) and "The Ancients of the Country." The Ancients were the people who lived to the west and south of the Natchez - notably for this discussion, the most remote of the Ancients "inhabited the entire coast of the great water which is toward the setting sun."

All in all, there's a widespread understanding in eastern North America at the time of European contact, that there are oceans to the east and west of the continent, and this understanding is based on some degree of exploration and experience, not just lucky guesses.