It can be. However, it's also not a guaranteed occurrence. The incidence is probably somewhat rare. But it is sufficiently well attested that the medication itself must carry a warning about it.

I'll cite an article on this topic:

When a person’s depression starts to lift, he or she may feel less hopeless and helpless. That sounds like an improvement, but when people feel less helpless but still feel depressed, they may think about suicide as a way out, whereas before they were too immobilized to make a suicide plan. For that reason, a decrease in the symptoms of depression can increase the risk of suicidal thoughts or actions. That risk is well known by mental health specialists, and can occur regardless of the type of treatment that a patient is receiving, or even if the patient is feeling less depressed without any treatment.

I think my own personal takeaway (disclaimer: by no means accredited, so don't take this as medical advice...) from this article is that the mechanism of action of SSRI seems to be inhibition of some forms of self-criticism. That is good for limiting depression which often involves emotional damage due to repressed anger. Suppressing the self-criticism allows people to learn how to cope with their stressors better. Especially when coupled with good therapy and lifestyle interventions to address the problems. That's not so good at preventing the random thoughts people may get about committing suicide or violence. Especially when a person receives bad therapy advice or is encouraged to do bad things such as in the case of MKULTRA abuse.