Transgenes are genes that have been transferred using genetic engineering.

Some say "Yes"...
Genetic changes are passed on to the next generation. This means that pollen from a GE plant can carry the transgene to non-GE types of the plant. This movement of DNA is called gene flow. Gene flow can be a problem for growers, such as organic farmers, who do not want to grow GE crops. Some plants have pollen that is blown by the wind or carried by bees. Transgenes can also spread to related wild plants.
Some say "No"...
First, cross-pollination can only happen with closely related plants. Gene flow can be controlled by spacing crops and timing when they are planted so that they can't cross-pollinate. The United States Department of Agriculture checks if cross-pollination with wild plants is likely to cause problems before GE plants are approved.

Learn more about gene flow:
Gene flow can occur if the related plants are close enough to each other, they bloom at the same time, and there is something to carry the pollen. Canola plants are an example of gene flow between GE plants and wild plants. Wild plants with the GE gene were found near canola fields and alongside roads. Canola has light pollen that is carried by the wind so it is not surpring that wild plants near fields would carry the genes. Canola has small seeds. Some of these seeds probably fell off of trucks, grew, and then pollinated the wild plants in the area along the road (source). There is concern that the GE genes for herbicide tolerance will spread to weeds, making them more difficult to control.
Pollen from GE plants is a problem for organic farmers who are trying to grow the same crop. Organic farmers are not allowed to grow genetically engineered foods and must create plans to avoid contamination with GE genes. Their crops must be blooming at different times than the GE crop or they must have enough space between fields. This can add to the costs of farming, especially for plants that have lightweight, wind-blown pollen.
The risk of gene flow is determined by the characteristics of the plant. Scientist look at the weight of the pollen and how long it stays alive. Plants like soy self-pollinate so the risk is low. In contrast, canola has pollen that is light and spreads easily.

Compromise?
Read this article on the approval of GE sugar beets.
U.S. Says Farmers May Grow Engineered Sugar Beets (New York Times, 2011)
Sugar beets store the sugar in their roots during the first year and flower during the second year. Typically, seeds are planted in the spring and the roots are harvested in the fall. Stalks are cut off of plants that start to flower early so they won't take energy from the roots.
New seeds are grown in mild climates where the plants can be sown in the fall, grow all winter, and then flower the following summer. There is concern that the pollen from the GE beets will spread to the fields of farms that are growing seeds for non-GE beets or other related crops. The 4-mile limit is based on how far the pollen can spread in the wind. (read more)
Even though the risk is low, you can't guarantee that no accidental cross-pollination with a non-GE variety will ever take place.
Do you think the 4 mile limit is a good compromise?