No. Bridge piercings pass through the loose skin above your bridge, not through the rigid part of your nose. (Actually, your nose holds its shape from cartilage, not bone—but it doesn’t go through that either.)

Since bridge piercings pass through the loose, flat skin above the bridge of your nose it is, essentially, a surface piercing. Unfortunately, this also means that it is prone to migration, rejection, and “growing out.” While there is nothing you can do to stop this from happening, a good aftercare regimen will help to slow the process.
If it does migrate or reject, remove it. Keeping it in longer will not help you save it and will only cause extra scarring. If you still want the piercing, hang on to your jewelry and get it repierced once the original piercing is closed and the area returns, both in look and feel, to normal.