Depends on the plant. Some have self-incompatibility mechanisms which make such things difficult. In many cases there's no problem, and in fact such self-fertilization is the norm.

Even in cases of self-incompatibility, there are means of overcoming it in some species.

Lilium species and hybrids are concidered self-incompatible. The pollen will geriminate on the plants own stigma, and start growing toward the ovules. Then, if it is compatible pollen, before it gets very far, the plant responds by producing something that makes the pollen tubes start growing faster. This means that pollen from other lilies will beat self pollen to the ovules. In fact, self pollen often doesn't make it in time before the flower withers. Keeping the lily in cool conditions helps the flower to remain in good shape until the self pollen does its job.
Walter