Yes, and no. As with any intelligence, it may be extremely difficult to raise one's emotional intelligence. Consider, for example the lessons learned from decades of early intervention programs. Certainly the popular press makes such a claim.
Many practitioners earn their livelihood by claiming that emotional intelligence can be increased.
Even if we accept the personality-based models of emotional intelligence, the notion that we can change, and easily change, basic personality traits just isn't correct. Certainly they cannot. Yes, this is possible. When it comes to an ability-based model of emotional intelligence, we need to proceed with great caution when we make claims about its malleability. There is no data that bears on this subject.