It is often thought that if one doesn't eat animal products, then they are at risk of not getting enough of the right protein. The idea of protein combining originated in 1971 as a way for vegetarians to make sure they were getting enough complete protein.  The theory goes something like this:

-animal foods are "complete" proteins (they contain all the essential amino acids)
-plant foods are "incomplete" proteins (they contain all the essential amino acids, but not in the same foods)
-therefore, plant foods must be combined in order for vegetarians to eat a "complete" protein at each meal.

This theory, started by Frances Lappe Moore's Diet for a Small Planet, gained huge popularity when it came out, but was unfortunately just a theory, unproven by science.  Since then, we've learned a few things.

First of all, there are some plant foods that are also "complete" proteins, such as soy, hemp, quinoa and amaranth.  Second, it just simply isn't necessary to make sure we eat complete proteins at every meal, since our bodies are smart enough to use proteins eaten at different times.

By now, the scientific community has discredited Moore's theory as being bunk. Even she has come out against it, saying,

"In 1971 I stressed protein complementarity because I assumed that the only way to get enough protein ... was to create a protein as usable by the body as animal protein. In combating the myth that meat is the only way to get high-quality protein, I reinforced another myth. I gave the impression that in order to get enough protein without meat, considerable care was needed in choosing foods. Actually, it is much easier than I thought."

Even conventional organizations such as the American Dietetic Association no longer tell vegetarians to combine proteins, since no one has been able to prove its necessity.  Many registered dietitians and medical doctors also agree that this view is long outdated, including Jeff Novick, John A. McDougall, and Charles Attwood.  Despite how old and disproved this myth is, though, some people and institutions still perpetuate the idea.

According to Jeff Novick, R.D., it is "virtually impossible to design a calorie-sufficient diet based on unprocessed whole natural plant foods that is deficient in any of the amino acids. (The only possible exception could be a diet based solely on fruit.)"  Though some plant foods are low in certain amino acids (for example, wheat is lower in lysine), they still contain enough of that amino acid to be more than sufficient for a person's daily needs. 

It can be said, then, that unless one is subsisting off of mainly fruit, which is low in protein and amino acids, or if one is eating a diet of refined and stripped junk food, or if one is subsisting off a single low-protein plant food (such as the cassava root in Africa), then they could be at risk or not meeting their protein and essential amino acid requirements.  For the rest of us, it is very simple to meet protein requirements for all of the amino acids - vegan, vegetarian or not.

It's important to combat myths such as the ever-enduring protein combining myth, since disinformation could cause people to eat more of the foods that are known to be harmful out of a concern for protein deficiency.  We already know that excessive animal product consumption is a major player in cancer, heart disease and diabetes.  Instead of scaring people into thinking they must eat meat based on old, unscientific viewpoints, let's support the good health of all by encouraging safe plant-based diets.

Diet for a Small Planet, Frances Lappe Moore, 1981 edition
Complementary Protein Myth Won't Go Away!, Jeff Novick, R.D., Healthy Times, May 2003
When Your Friends Ask: "Where Do You Get Your Protein", McDougall Newsletter, John A. McDougall, April 2007