Requirement of logarithmic divergence basically is result of local nature of counterterms and renormalizability of the Lagrangian of the theory however there are some exceptions. If you can have a local counterterm (product of fields or their derivatives) which has already a form similar to one already present in the Lagrangian then it is possible to redefine some parameters to absorb the divergence.
In general, a divergence higher than logaritmic can destroy renormalizability of the theory and to make a better sense of it one has to use an explicit cut-off making it an effective theory upto that cut-off.
In case of QED, suppose you encounter a quadratic divergence for self-energy of the electron; this would require to introduce a term like $C_1(\partial_\mu\psi)^2$ as $\partial_\mu$ gives a mass dimension one and does not introduce an explicit cut-off scale of energy. In QED, we don't have a term like $(\partial_\mu\psi)^2$ as electron field introduced only comes with first order derivative term in Lagrangian. As a result, it is not possible to absorb the counterterm with any other existing term. This process is basically an indication of non-renormalizability of the term which can be seen simply also from counting the mass dimension of the term. A logarithmic divergence on the other hand does not introduce any extra mass dimension to a counterterm so these problems do not arise.
Another case of Higgs field is interesting as the quadratic divergences do not spoil renormalizability. A quadratic divergence requires a counterterm of the same form $B_1(\partial_\mu\phi)^2$ but now this has the same form as the kinetic term of the scalar field in the Lagrangian. So we have a term of the same form in the Lagrangian already and hence it's not problematic on the general ground . Power divergences however cannot be handled by renormalizable theories (without using a cut-off obviously); these theories are termed unnatural and other extensions are needed to treat it properly (like SUSY, Technicolor). Without these extensions one have to fine-tune the theory suffering from power divergence.
This discussion is parallel to Collins 'Renormalization' chapter 3.