How Deliverances Can Fail
The most common obstacles
Not enough faith - Jesus said that some demons only come out after prayer and fasting, which builds your faith to the higher levels required to cast out some of the stronger demons.
Mark 9:29, "And he said unto them, This kind can come forth by nothing, but by prayer and fasting."
Unwillingness to be delivered or let go - if the person is not willing to give up certain things in their life that are giving the enemy the legal right to remain, or they are not willing to let go of certain demons, there's little you can do to drive out those demons.
Legal rights were not broken - if you try to go about a deliverance without first breaking off any legal grounds given to the enemy, you will almost certainly fail, and even if you are successful, the demons will almost certainly come back with even more demons, making a worse mess then before.
Important steps were not taken - there are some steps that can make a day and night difference in a deliverance session, such as getting ahold of the demons names and commanding them by name. If you simply command all the demons present to leave, you will likely be ignored; it's like somebody trying to get your attention in a crowd by yelling "Hey you!" Many times when Jesus cast out demons, He referred to them by name, and He got their attention in quick order. Not every deliverance session requires you to refer to the demons by name, but it sure helps when you are able to get ahold of their names, either through the person your ministering to (if they accepted a spirit guide by the name of Daemon, then they would obviously know the name of one or more spirits that are in them), another way to get the name is by the symptoms they are causing, for example, extreme guilt feelings are often caused by a spirit by the name of guilt, and other ways to get their names are by a word of knowledge from the Holy Spirit, or from the demons themselves (if they manifest, ask them for their name).
The person hasn't accepted Christ - trying to cast a demon out of an unbeliever is like trying to kick somebody out of their apartment when their rent is paid up; it isn't an easy task and even if you did succeed, they are bound to come back with several more even worse evil spirits as Jesus warned in Matthew 12:43-45. This problem can be easily solved by asking the person if they would accept Jesus, with a mouth confession as the Bible says, and believe with their heart in the Lord Jesus Christ.
Romans 10:9-11, "That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved. For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation. For the scripture saith, Whosoever believeth on him shall not be ashamed."