Probably not, although the question is more complicated than it seems.
Part of the confusion stems from kindergarten being a foreign word importation. The word was coined by Friedrich Fröbel, a German educator who founded a school for young children in the 19th-century. All nouns are capitalized in German, and Kindergarten was no exception.
There is some inconsistency over whether German importations into English should retain their capitalization. For words that have become extremely common, such as bratwurst or angst, the answer seems to be no. However Schadenfreude, an uncommon but still familiar word, is capitalized in the Oxford English Dictionary.
It is difficult to determine at exactly what point a German importation reaches a level of usage sufficient to drop the capitalization. Of course since kindergarten is a very familiar word to most English-speakers, we could assume that lowercase is correct.
But it's not as simple as that. Style guides from some authorities, such as the International Reading Association, have in the past instructed writers to capitalize all grade levels, such as First Grade, High School, etc. Kindergarten would be subject to that rule. While the IRA guide has departed from that policy, the confusion lingers.
Complicating things even more are those who insist that references to a particular kindergarten program be capitalized, as in Springfield Elementary Kindergarten or Ms. Johnson's Kindergarten. There is also the issue of abbreviation, as in K-5 or pre-K. In those cases, the capital K reflects the convention of capitalizing abbreviations, not the convention of capitalizing the word itself.
Taking all of that into account gives us several rules for different cases. If you are referring to kindergarten in general (and in English), you may use the lowercase. Specific references to a single kindergarten program may require capitalization, and you should use a capital K in abbreviations.
Yes, "Pterodactyl" is supposed to be capitalized as it is a proper noun referring to a specific genus of pterosaur.
Yes, the 'T' in 'The Bahamas' is supposed to be capitalized because "The" is part of the country's official name.
No, "adults" is not typically capitalized unless it is part of a proper noun or at the beginning of a sentence.
Yes, "Hindenburg" should be capitalized as it is a proper noun referring to the German airship LZ 129 Hindenburg.
Yes, "Arabs" should be capitalized when referring to the ethnic group or people from the Arab world.
Earth, as in we live on the planet Earth, is supposed to be capitalized. Earth, as in the dirt we walk on, is not supposed to be capitalized.
Usually, yes a phase of the moon is supposed to be capitalized.
No, it shouldn't be capitalized.
No it shouldn't be capitalized.
It is capitalized at the beginning of the sentence or when it forms part of the proper noun. Examples: Cinderella Pre-kindergarten School Pink Ballerina Pre-kindergarten Center
Yes, America should be capitalized.
no
Yes, "Pterodactyl" is supposed to be capitalized as it is a proper noun referring to a specific genus of pterosaur.
No, but the names of individual languages are capitalized.
Yes, Siamese would be capitalized.
Only if it's part of the name of the school.
Yes.