I found these two cancellation theorems in the K-book, in section 1.4, page 39:
Real Cancellation Theorem: Suppose $X$ is a $d$-dimensional CW complex and $\eta:E\to X$ is an $n$-dimensional real vector bundle with $n > d$, then:
(i) $E\cong E_0\oplus T^{n-d}$ for some $d$-dimensional vector bundle $E_0$.
(ii) If $F$ is another bundle and $E\oplus T^k\cong F\oplus T^k$, then $E\cong F$.
Complex Cancellation Theorem: Suppose $X$ is a $d$-dimensional CW complex and that $\eta:E\to X$ is a complex vector bundle with $\dim(E)\ge \frac{d}{2}$, then:
(i) $E\cong E_0\oplus T^k$ for some vector bundle $E_0$ of dimension $\le \frac{d}{2}$.
(ii) If $F$ is another bundle and $E\oplus T^k\cong F\oplus T^k$, then $E\cong F$.
I am interested in understanding the subtlety that appeared in the statements of these theorems: it seems like the complex case is analogous to the real case with half of the dimension since $\mathbb C\cong \mathbb R^2$, but the dimensions we get to pick are somewhat different. I tried to look up the reference in the K-book, which is Husemoller's Fibre Bundles, but I did not find what I want. Can anyone provide a reference or give an explanation? Thanks.
These two theorems follow from obstruction theory, see here for example. Let me address why the inequalities are different in the real and complex cases. I will instead denote the real/complex dimension of the fibers of a vector bundle $E$ by $\operatorname{rank}_{\mathbb{R}}E$ and $\operatorname{rank}_{\mathbb{C}}E$ respectively.
As discussed in the linked answer, if the rank of $E$ (as a real vector bundle) is greater than the dimension of $X$, then $E \to X$ admits a nowhere-zero section $s$. This spans a one-dimensional trivial subbundle of $E$ (just as a non-zero vector in a real vector space spans a one-dimensional subspace isomorphic to $\mathbb{R}$), so $E \cong E_0\oplus T$ where $\operatorname{rank}_{\mathbb{R}} E_0 = \operatorname{rank}_{\mathbb{R}} E - 1$ and $T$ is the trivial real line bundle. Provided $\operatorname{rank}_{\mathbb{R}}E_0 > d$, we can repeat the same argument - this yields (i) in the real cancellation theorem.
Suppose now that $E$ is a complex vector bundle. Since $E$ is also a real vector bundle, we still have a nowhere-zero section $s$. Now however, it spans a one-dimensional trivial complex subbundle of $E$ (just as a non-zero vector in a complex vector space spans a one-dimensional subspace isomorphic to $\mathbb{C}$), so $E \cong E_0\oplus T_{\mathbb{C}}$ where $\operatorname{rank}_{\mathbb{C}}E_0 = \operatorname{rank}_{\mathbb{C}}E - 1$ and $T_{\mathbb{C}}$ is the trivial complex line bundle. Provided $\operatorname{rank}_{\color{red}{\mathbb{R}}}E_0 > d$, we can repeat the same argument. Since $\operatorname{rank}_{\mathbb{R}}E_0 = 2\operatorname{rank}_{\mathbb{C}}E_0$, the requirement $\operatorname{rank}_{\mathbb{R}}E_0 > d$ is equivalent to $\operatorname{rank}_{\mathbb{C}}E_0 > \frac{d}{2}$. This yields (i) in the complex cancellation theorem.