我感觉自己生病了
come down with something是美国俚语,有“生病,不舒服,染疾”的意思
come down with something to become ill with a disease
e.g Twenty-two pupils came down with flu.
（22个小学生染上流感）
以上出自《剑桥字典之美国俚语》-Cambridge Dictionary of American Idioms
另外
come down 表示“下来”的意思
e.g I expect God to come down and erase all the worries from my mind.
我希望上帝能够降临到我身边帮我消除所有的忧虑
come down to (something)：
to be recognized as the most important thing.
意思是“被认为是最重要的事”
e.g It comes down to simply teaching the basics to these kids--they don't know anything.
对这些什么都还不懂的孩子来说最重要的是简单地教他们最基本的东西