This occurrence appears to happen when the speaker is trying to sound cute, or is possibly under the influence of something... They break one of the words into syllables or characters, with pauses in between where they broke up the word.
For an archetypal example:
"Do you want dinner? Or do you want a bath? Or do you want me?"
Or, when one character confesses their love to another, you might hear だ・い・す・き da-i-su-ki "I-love-you".