It is well known amongst the bonsai fraternity that young trees require growing on to develop trunks with the characteristics necessary for use a bonsai.
Accelerating the growth of young trees helps build thick trunks and good root spread (nebari) prior to planting the tree into a bonsai pot for refinement.
The simplest and most commonly used method of trunk thickening is to plant the tree in the ground for a number of years. Unrestricted rootgrowth in a large area of soil helps promote the strong and vigorous top growth that in turn thickens the trunk.

There are times that growing on in the ground cannot be carried out for a number of reasons.
Available land is not always available and some species require frost protection during the Winter and there is a need to be able to move them from their Summer position.

The simple solution would seem to be to develop the tree in a pot. Though speed of growth is reduced in comparison to field growing; good results can still be achieved this way.
However, the size of the pot used for growing on has a major influence that determines the speed at which the tree grows.

Underpotting

A severely rootbound tree is one that's roots have filled the pot to the extent that it has little or no soil for new roots to grow into; it will survive and issue new leaves in Spring but barely develop new shoots.
This lack of growth will make trunk development and thickening non-existent. This situation is remedied and can be avoided by ensuring the plant is in a pot with sufficient room to extend its roots. (A developed bonsai will of course be rootpruned so it can be kept in a small pot whilst still having fresh soil introduced).

Overpotting

The fear that growth will be slowed in a small pot leads many bonsai enthusiasts to overpot and advise overpotting.
Many times I have seen and heard a well-meaning enthusiast advise planting a young tree into as large a pot as possible "to speed growth".
Unfortunately planting a tree into too large a pot/overpotting is as detrimental to vigorous growth as underpotting and leaving a tree rootbound.
It has to be understood that planting a tree into a large pot is not the same as planting into the ground.

The following is an excerpt from a thread on the The BonsaiSite forums and is the best explanation of under/over potting issues I have seen. Nurseryman Brent Walston of Evergreen Gardenworks explains some of the reasons for regular potting on as opposed to overpotting.

"Back to the physics for a moment. Water will drain from a pot until the lowest level of saturated soil (that can be supported) is reached. At this point drainage stops and this saturated layer remains saturated, no more water will drain out (ever). The height of this column of soil depends on the nature of the mix. A coarser soil will have a lower (shallow) column or layer of saturated soil than a finer mix. The total retained amount of water is less for a coarser soil.

Water can be removed from this saturated layer. It can be removed by evaporation (the water will be wicked upward as water evaporates from the surface), or it can be wicked out by the absorption of water by the roots (powered by foliage transpiration)."

"If the plant is not root established, it cannot remove very much water by transpiration. This leaves too much water in the lower levels of the soil. In the short run, this is not much of a problem."

"However, and this is where the problem is, IF the pot is so large that the saturated level cannot be removed by normal root colonization, then problems begin."

"What happens if the limits are exceeded? If you are using an organic amendment such as bark, you will experience accelerated soil composting. This means that you will lose your effective soil particle size more quickly than if you used a smaller pot which is wicked dry daily. This is the most common effect. You use a pot that is too large and stays too wet. The organic amendment quickly decays in this wet environment, particle size decreases, soil collapses, saturated level increases, even more water is retained, roots eventually remain in standing water, root failure occurs with or without the presence of a pathogen."

"Even if the above doesn't occur, what kind of root growth occurs in a volume that is not wicked dry daily? When you water properly, a new charge of air is pulled into the pot by the volume of water draining from the drain holes. CO2 and other gases are purged from the soil. The longer you leave these gases in the soil, and the longer you wait to introduce a fresh charge of oxygen, the poorer the roots will be. If you create a situation such as over potting that doesn't require daily watering, then you don't obtain optimal soil growing environment.

The BEST environment is a soil that dries out daily. The best potting practice is to shift to the next larger size pot after each time the plant becomes root established as evidenced by forming an intact rootball. UC Davis studies have proven this, and I have conducted my own studies with Acer palmatum which have verified it to my own satisfaction. It is not a marginal effect, the resulting growth improvement is significant."

"The best way to achieve fastest growth is to shift (repot) just as soon as the plant produces an intact rootball. This is standard nursery practice and a well established principle. If you do this, you don't have to disturb the rootball or prune the top, thus there is little or no shock and it can be done at any time of the year. Bonsai practices somewhat complicate this, since we want specific root configurations, but for plants in training it still holds.

Ok, what's an intact rootball? An intact rootball is when you can knock the nursery can or pot off the root ball and it won't fall apart."

"Even after a plant 'apparently' occupies all the soil spaces with roots, it may still grow normally for some period of time. This is probably due to two factors that I can think of. One is that tiny hair roots are still growing, exchanging gases, absorbing nutrients, etc. Secondly, the somewhat larger roots are not yet 'lignified', or woody, and thus are still also fairly active.

I think it is better to determine 'rootbound' by both the symptoms of growth (or lack thereof) and the physical density of the roots. For our purposes (bonsai), trees should be rootpruned and repotted LONG before they reach rootbound conditions. This doesn't happen overnight. There is a long gradual procession of slowing growth over time, usually several years before all new growth stops. It is clearly evident what is happening if you stop to look."

As a conclusion; I would strongly advise regularly potting on trees into larger and larger pots as and when the root mass demands it.
Whilst it is good practice to find the right size pot for a particular tree the exact size is not absolutely essential; just don't be fooled into using greatly oversized containers and occasionally check that your trees in development (potentsai) have not become rootbound.