There are basically three different concepts that are related in this context. Strength, endurance and stamina. Many people use stamina and endurance as synonyms, but they have distinct differences.
- Strength - The amount of weight/force your muscles can withstand or produce.
- Stamina - The amount of time that your muscles can perform at maximum capacity
- Endurance - The amount of time your muscles can perform a specific action.
So, if you are all out sprinting, and you can last 15 seconds at that effort and your buddy can last 20, he has more stamina. If you are both jogging, and you can go an hour while he can only go 30 minutes (at the same pacing), you have more endurance.
For your dancing, if you do nothing different, you can increase the time you can spend dancing by decreasing your intensity (Obviously not ideal for a competition). The other way to dance at the same level longer is by increasing your endurance, since you are not performing at 100% all out in competition. This will have a few effects, your body will become more efficient at utilizing glycogen, your body will store more, your lungs and muscles will become more efficient, and your energy cost at a given level will decrease.
As you note, you can increase this time by practicing more/longer, or you can add other endurance training (Swimming, running, rowing, cycling, xcountry skiing, etc). The endurance effects from one will cross over to the other.
This is a simple explanation of a fairly complex topic, there are quite a few different changes that happen with energy pathways, blood vessel growth, oxygen efficiency, mitochondrial changes, etc.
Also, to answer your base question, yes, there is a maximum amount of stamina and endurance you can have, and it is limited to your readily available energy and your stored energy. Without adding any external source of potential energy, it is a finite resource that will run out.
At a maximal effort, you will most likely be exercising anaerobically (without oxygen), so you are limited to at absolute most 30 seconds, also limited by available ATP in the cells. Since the Krebs cycle will replenish ATP in the presence of O2, then you are limited to body stores of glucose/glycogen and in extreme cases, fat/muscle conversion.