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The short answer is yes, you would have completed 28 years of life at 11:55 am on January 14, 2016 (Happy birthday, if that is your actual birthdate!).
The longer answer depends on how picky you wish to be about things like leap years, leap seconds, and how one wishes to define one year.
One definition of a "year" is 365 periods of 24 hours. If one wanted to be strict about this, you would have completed 28 full "years" a week prior to January 14th, due to the seven leap years that have occurred between January 14, 1988 and January 14, 2016.
Another definition of a "year" is the time that it takes for the Earth to complete one revolution of the Sun and return to its "starting point" in its orbit. This period is approximately equal to 365.25 days. After four years that additional 0.25 days per year adds up to a full day, which is why we have a Leap Day every four years (with some exceptions).
There are also occasional "leap seconds" added to make small adjustments to the current time.
Thus, the "exact" time depends on how one defines a year, and how one wishes to treat the various adjustments.
From a civil/legal perspective though, you would have been considered to turn 28 at the stroke of midnight between the 13th and the 14th of January, without regard to the exact time of your birth on that day, or any of the various adjustments made to the calendar during those years.
I hope this helps. Please feel free to ask if you have any additional questions about this.