Can a family of six have six service animals in an apartment?

Yes, you may have two service animals, provided they each assist in your disability. For example a blind individual may have a guide dog for their every day use, however may also have a severe diabetic condition where a smaller dog is able to alert. In this instances both dogs would be required since each does a different job that the other can't do.

Though it's legal to have two service dogs you can expect resistance while traveling since it's very uncommon.