There are situations in which vast quantities of IV heparin fail to increase the APTT in spite of your every effort. One might call this "heparin resistance", or "heparin insensitivity". For some reason, this issue seems to enjoy a significant amount of attention from the college. Particularly, Question 13.2 from the first paper of 2012, Question 8.1 from the first paper of 2011 and Question 6.2 from the second paper of 2008 all ask the candidate to give reasons as to why the patients APTT might be so low in spite of heparin therapy.
There are several reasons one might be resistant to heparin:
UpToDate offers a good article about Antithrombin III deficiency. Either you hereditarily fail to synthesise enough of it, or your liver is so damaged that it cannot produce enough. Or, it has been used up somehow, eg. in the context of DIC, MAHA, or in a bypass circuit. Lastly, it is possible that you are losing it along with other proteins via your leaky nephrotic kidneys.
The management of AT-III deficiency is, predictably, supplementation with AT-III.
If the expensive purified factor is not available, FFP will suffice.
There are several strategies one can employ. The specific choice relies on what exactly is causing the heparin resistance.
There are some good articles on this. Most of them do not touch upon the routne anticoagulation of some random patient who happens to have escalating doses of heparin; I suppose it is generally assumed that one will continue to escalate the dose until such time as therapetic goals are met. However, there are situations when anticoagulation is critically important, and one such scenario is the cardiopulmonary bypass circuit.
Or, you could consider using something else, such as a direct thrombin inhibitor (hirudin or argobatran)