The best time to transplant your tropical plants is in the spring time. This is when they are just starting to produce lots of new growth. This means the roots will be more able to re-establish before the dormant season in the fall. It is still ok to transplant during the growing season as well. However, you should try to avoid transplanting during the winter season unless it is absolutely necessary.
The key is recognizing when your plants are root-bound in their containers.
There is one final check to see if your plant is root-bound:
If any of the 3 above signs are present that may be a sign that your plant is need of re-potting. The final check is to remove the plant and examine the roots. If you have a matted root mass then you need to re-pot, preferably sooner than later.
Consider transplanting your house plants into larger plastic growers' pots and then you can use ceramic containers to hide the plastic. This makes watering easier if the plant needs to be transported to the sink or tub. A wide range of decorative containers are available in many styles for all settings.
Plants that already have very large pots and containers...
Larger specimen plants that have larger pots or containers like to be more root-bound. At this stage in their life they won't be growing as much as when they were much smaller and younger.
Even though they don't need to be transplanted they still need nutrients. Fertilizing solves this problem although after several years with nothing but chemical fertilizers the soil is by best definition; dead! Top dressing is the answer here. Simply remove the top few inches of soil around the plant and replace with a fresh potting soil or compost. This will add much needed micro organisms back to the soil and provide your plant a healthier living environment. Continue your regular fertilizer schedule as normal and consider starting one if you don't already.
Above: A very, very root bound Monstera. Notice the roots have even engulfed the label for the plant.
Above: A large palm in a growers container hiding in a decorative planter. Much easier to manage larger plants in this manner. The tin-foil balls help keep your kitties from using plants as a bathroom.