This question depends on the acceptance/non-acceptance of an objective reality that remains constant. If we assume that reality is objective (A is A no matter what we think or believe) than it can be said that perception does not always reflect reality. Certainly this can be seen in personal bias, which affects perception regardless of reality. Many people believe that their personal perception is a "personal reality" unique to themselves and barred from others; this is somewhat, but not entirely, disproved by the fact that so many people can share the same objective experience.
Going deeper, a person with mental issues such as hallucinations will not experience the same reality as a person without those issues. The sufferer's reality will necessarily be "different" but no less "real" to their personal perception. If we accept an objective reality, we can then accept that alternate perceptions are not "reality" and so are explicitly personal.
In this sense, then, it becomes not necessarily the perception itself that is biased, but the interpretation of that perception. Each individual interprets their experiences based on their knowledge and history of past experience; each individual then applies those aspects to their perceptions and so comes up with an "individualized reality." This allows multiple interpretations of the same experience; a joyous event for one might be a tragedy for another. Perception becomes a tool used in personal interpretation; assuming an objective reality, perception cannot be "the same" as reality, but just a method of personally interpreting what is already there.
Yeah, I always believe in this phrase that perception is reality. It is really so true because I think what people distinguish is usually what they believe and it is based on what they perceive, hear and think.