Short answer. Usually not. A witness has the right not to give an answer that will tend to degrade his or her reputation. But the protection falls away where the degrading matter is the very fact at issue, and a witness must always answer as to a previous final conviction.
What the law says
Not to give an answer which will tend to degrade his or her reputation, unless it be to the very fact at issue or to a fact from which the fact in issue would be presumed.
Rule 132, Section 3 — Rights and obligations of a witness. Read the full provision →
The starting position favours the witness
The rule opens by saying a witness must answer questions even where the answer may tend to establish a claim against them — testifying is not optional and discomfort alone is no excuse. It then lists rights that qualify that duty, and one of them is the right not to give an answer that will tend to degrade the witness's reputation. So a question aimed at exposing something shameful about you, rather than at the matter the case is about, is one the rule already contemplates you may decline.
The exception is where most of these questions are decided
The protection is not unconditional. It does not apply where the degrading matter is the very fact at issue, or a fact from which the fact in issue would be presumed. The reason is straightforward: a court cannot try a case while the central allegation is off limits because answering it would look bad for the witness. Where the very thing in dispute is the conduct you would rather not discuss, the rule does not let the protection swallow the case. The line therefore turns on how close the question sits to what is actually being decided.
A previous conviction is carved out expressly
The rule adds a specific instruction that leaves no room to argue: a witness must answer to the fact of his or her previous final conviction for an offense. Notice how narrowly that is drawn. It is the fact of a previous final conviction that must be admitted. A witness cannot refuse that question on the footing that admitting it is humiliating, because the rule has already weighed that and decided the other way.
How it works in the courtroom
This is a right, not a self-executing shield, and the rule places it alongside the witness's other protections — from irrelevant, improper or insulting questions, and from being examined on matters not pertinent to the issue. In practice a witness is rarely well placed to judge on the spot whether a question is aimed at the fact in issue or merely at their character. If you are due to testify and expect questions about matters that would embarrass you, raise it with counsel beforehand rather than deciding alone in the witness box.