Sure, Ti can rust. Rust is properly called "oxidation."
When Iron reacts with Oxygen, it forms Iron Oxide which is a redish to grey-blackish flakey powdery substance. That's what we normally think of as rust. This reaction is slow.
When Aluminum reacts with Oxygen, it forms Aluminum Oxide which is a crystal-clear substance that is in the same league as diamond for hardness. Aluminum Oxide is commonly used as an industrial abrasive. This reaction is very, very fast. Virtually every piece of aluminum you've ever seen has been completely covered with rust. But, because it's clear, you don't see it. And because it's so tough, it actually self-limits. After a layer only a few molecules thick forms on the surface, no more oxygen can get in and the reaction stops. So, the reason Aluminum is commonly thought of as rust-proof is that it's already covered in rust.
When Titanium reacts with Oxygen, if forms Titanium Oxide which is a powdery white substance that's fairly soft (on the same order as chalk). Titanium Oxide is very, very bright white. In fact, it's a common pigment in white paint. It's also common in cosmetics, especially powders where it lightens the color, gives a bit of a "sparkle" appearance that people associate with purity and cleanliness, and also helps keep the powder from caking and clumping. The oxidation of Titanium at room temperature proceeds at almost glacial rates. So, your cup will rust, yes, sorry, but not in your lifetime or that of your children or grand children or great-great grandchildren or...
The only thing I can think of is that your kettle is made of a titanium alloy (amalgamation of 2 or more metals) and one or more of the alloy components is inferior, causing it to rust like steel etc