Different orchids have different growth behaviors.

Monopodial orchids like a vanda and phalaenopsis will continue to grow from its crown and for the most part, maintain the appearance of a single plant. Flower spikes will emerge from segment nodes along the length of its body.

Cattleyas and oncidiums belong to the sympodial orchids. And even in the same category these two have different growth behaviors.
The cattleya pseudobulb would have the flower spike emerge from the crown. This is the segment where the bottom of the leaves and the top of the pseudobub meet. Once a cattleya has bloomed (or the flower bud is destroyed) the pseudobulb cannot bloom again.
A new pseudobulb would emerge from a foliar bud (also called "eye") near the bottom of the pseudobulb. This will mature and then have the ability to bloom and eventually produce its own new pseudobulb. Depending on growing conditions and parentage of the cattleya, one or two new pseudobulbs could emerge simultaneously.

Oncidiums mostly grow flowers from the base of the new pseudobulb once it has matured. The new pseudobulbs also emerge from foliar buds near the bottom of the current pseudobulb.

As to your question about specimen plants: Those plants have matured enough so that they produce multiple new pseudobulbs every growing season. Since each pseudobulb has the potential to bloom, the specimen plants can produce many flower spikes in one growing season.

Here is a simplified example.
If a cattleya pseudobulb reliably produces two new pseudobulbs every growing season, and if you started with just one pseudobulb.
Then after 5 seasons you will get:
1 x 2 x 2 x 2 x 2 x 2 = 32 new pseudobulbs that could bloom.
This is not counting all the old pseudobulbs.

But of course, expecting two new growths from a cattleya pseudobulb year after year might be unrealistic for most of us.

Cattleyas are indeed sympodial orchids consisting of a rhizome and a pseudobulb(s) (notice the word pseudo, i.e. 'false' because it isn't really a bulb like a tulip or a corm like an onion). At any rate, cattleyas bloom on new growth. A matured pseudobulb. With some species (or with a certain species in its background) a cattleya will bloom on the newest matured growth like C. warscewiczii (gigas). In the case of C. mossiae or trainae, new growth is formed, the psuedobulb matures and then it will bloom in the winter (trianae) and mossiae spring. Cattleyas have two main "eyes" at the base of the pseudobulb. Many times, the plant may only sprout one eye. Or it can sprout both. There are tiny backup eyes just above the main eyes. These are emergency eyes. They will usually remain dormant and never grow. I now spray each dormant eyes with a solution of Keiki Grow to stimulate these dormant eyes. I've found that it works.

Even though the pseudobulb will never bloom again, it continues to provide strength to the plant, so it should not be removed until it dries up and turns brown.

Thanks so much for the responces! i was feeling a little confused about it, but what i thought might be true is proven, but at the same time it is dissapointing knowing that only one pseduobulb will make some flowers and then not make anymore, but how many flowers can we expect a group of plants to make each year?. i do have some questions though on oncidiums/miltonias, i currently have one plant and i am wondering do they flower each time they make new growths- sheddings of leaves when they make their bulbs bigger, or are they like cattleyas. i appreciate all your help, and answers.

Pseudobulbs start to multiply on the plant, and very often two or three bulbs will produce flowers at one time, so a plant can have many flowers as it matures.

The fact that an "old" unflowered psuedobulb will not bloom isn't really a dissapointment. (Though, many times, what looks like unbloomed pseudobulb, i.e. without a sheath, will flower. Again, the bulb must be rather new). But a spent pbulb will not flower. But again, the potential for more flowres exist in each new eye that "pops." The amount of flowers per inflorescence is goverened by its genetic past. Some many only put out 1 or 2 blooms (Rhyn. digbyana reduces flower count because it only produces 1 or 2 flowers) or C. gigas which can put out 5 or more huge flowers. And then factor in hybrids with mulit-flora habits. A well grown specimen plant can literally be "covered" in blooms. But that to happen, the plant needs optimal light, food and other environmental conditions. Light is the biggest factor.

My Rhyncholaelia Digbyana currently has 3 blooms, and I have seen pictures of it with 5-7. Last year mine had two promising buds, but they dried up, perhaps due to insufficient light. I am still hoping some day those dry sheaths will product something.
Ann

I have two orchids, one bloomed this past year; however, the other did not. Both are healthy looking, but my main question is, what do I do with all of the long, root-like projections extending out from the plants...do they bloom or provide anything for the plants?