In Mexico's hot and humid Yucatan peninsula are many examples of exactly what you are asking about, some dating back to 400 BC.
Many are abandoned in the middle of the rain forest, while others have been tidied up for tourists.
The most famous is "El Castillo, Chichen Itza", which has been declared a UNESCO World Heritage Site:
This one is only a thousand years old, but it could obviously continue surviving for many more centuries or millennia.
We can build structures that will last for thousands of years, but we generally choose not to.
EDIT: How is it able to not fall apart like our structures? We construct a concrete foundation, its cracking almost immediately. – J. M. Becker
The Coliseum in Rome was built over 2000 years ago, and most of it is still standing. Much of it is made from concrete that has endured all this time. The Romans expected it to last "forever".
Consider the equivalent of the anthropic principle. The ancient societies did build many flimsy structures, but the only examples that remain of their work for us to see are the few that were well built.
But there's another factor to it. Today's society is very different from what it used to be.
The Mayans built their structures as permanent monuments. They were intended to last. (Egyptian pyramids are nearly 5000 years old, but the original question asked for a "humid climate", so I didn't use them as an example.)
Today's structures are built to make a profit for their builders, and to provide a quick and modern building for their occupants. In a few years they will be old fashioned and replaced by another trendy building. There's no point in spending a fortune building something that will endure.
Our current society and economy is based on this. I use tools that were made 50 to 100 years ago, and they are still in good condition. Today, unless I'm willing to spend an incredible price for high quality manufacturing, anything I buy now will be cheaply made in China and will need to be replaced in a year or two.
In today's dollars, years ago I could buy a tool for \$20–30, depending upon the quality I wanted. Today I can buy a tool of similar quality for \$150, or a cheap version for $5. There's nothing in between. Most people choose the \$5 version, which means that the good quality versions must be made in small quantities, and so end up costing much much more.
Household refrigerators that were made 50 years ago are still running fine (energy inefficient though), while most made 10 years ago are now scrap.
When it comes to houses and office buildings, the situation isn't much different.