Hi. I think you can have an infinite number of consecutive ands and still create a grammatically correct and meaningful sentence. Here is how it works. Assume that you are not pleased with the spaces put between the words "Pig and And and And and Whistle" in the sentence "I want more space between Pig aNd And anD And aNd Whistle" (Capitalization used to mark the different ands.)
You may say "I want more space between Pig and aNd, aNd and And, And and anD, anD and And, and aNd and Whistle" (I have used the "Oxford Comma" for clarity.)

Then if you are not happy with the spaces between the words "Pig and aNd, aNd and And, And and anD, anD and And, and aNd and Whistle" in that sentence you can write another sentance which has even more ands, and you can keep going on and on in this way.