The short answer is yes. However, the number of times you can reload each cartridge varies depending on a few different factors.
Reloading depends on the integrity of the bullet or shot casing. Each time it is shot, the lifespan of a bullet drops. When the metal is worn enough, it cracks and separates. You want to stop using the case long before a critical breakdown like that.
In most cases, a rifle case should be discarded after five to seven uses. Many shooters simply toss their cases after the fifth reload without even taking the time to inspect it because five is generally the limit for necked rifle cases. It is very difficult to inspect the bottom of these long cases for head separation that may not be noticeable from the outside and it is just not worth the risk.
Pistol cases, on the other hand, are much easier to inspect and can be used until you see a case mouth split. Therefore, a pistol case can be reloaded anywhere between seven to 15 times.
The number of times you can reload a cartridge ultimately depends on the case and the load that you use. Annealing and thinning of the case are two major factors in the life of the case. Usually, cases are fire-annealed and hardened in specific areas in order to make them flexible or stiff in the right ways. Firing and working the cases repeatedly will eventually lead to poor performance when it comes to seating, resizing and fitting into the chamber properly. This is the reason it is never a good idea to dry cases in the oven after you wash them.
When you fire and resize a case, it stretches and becomes thinner. The cases have to then be gauged and trimmed. This extra metal has to come from somewhere, usually creating a weak point near the base of the cartridge. Upon close inspection, you will be able to tell when a case is likely to fail and should not be reloaded. The area close to the base will appear stretched, similar to a plastic bag.
When you reload ammunition, it is a good idea to do it in lots, keeping track of the number of times you have reloaded each lot. In addition, you should inspect the cases every time to check for any damage. You might also want to record the case length before and after you resize it.
Once you have trimmed your cases two to three times, you should not reload it anymore. If you have a band saw, you can split a few cases down the middle to see the thickness of the walls and educate yourself further about what happens each time you reload a round.