Simultaneous Confidence Interval 
I don't understand the example here. 
If we have that $\alpha=0.04$, $k=2$ then $[L_1,U_1]$ is a $1-\frac{0.04}{2}=0.98$ confidence interval for $\theta_1$ which is correct, however $[L_2,U_2]$ must be  a $1-\frac{0.04}{2}=0.98$ confidence interval for $\theta_2$, which is incorrect as $0.98>0.97$

Each individual CI has its own confidence level (and hence error
probability). When considering the error probability arising
from several CIs considered simultaneously, the error probabilties
can "gang up" to be larger than the error probability of any one CI.
What you are seeking here is a 'family' error rate.
The explanation in the NIST handbook may be helpful. This
method is based on 'Bonferroni's Inequality' and may slightly
overestimate the family error rate.
In addition to the very general Bonferroni method, other methods
of finding a family error rate (especially in ANOVA) have been
proposed: 'Tukey Honest Significant Difference (HSD)', Scheffe.
method, the method of 'Student-Newnan-Keuls' (SNK), and several methods by Duncan.
