Is the amount of cholesterol the body produces affected by how much cholesterol you take in with food?
Here's a study in which they measured a decrease in endogenous cholesterol synthesis after various amounts of dietary cholesterol: Dietary Cholesterol Feeding Suppresses Human Cholesterol Synthesis Measured by Deuterium Incorporation and Urinary Mevalonic Acid Levels (Arteriosclerosis, Thrombosis, and Vascular Biology, 1996):


  The objective of this study was to measure the response of cholesterol
  biosynthesis in subjects to three different amounts of dietary
  cholesterol: 50 (low), 350 (medium), and 650 (high) mg cholesterol per
  2800 kcal. Individuals with low (n=7), normal (n=12), and elevated
  (n=11) plasma cholesterol concentrations consumed in random order
  solid-food test diets (15%, 55%, and 30% of energy as protein,
  carbohydrate, and fat, respectively) at each dietary cholesterol
  level. The three diets were consumed for 4 weeks each, and each
  dietary phase was separated by a 4-week washout period. During the
  final week of each diet, 0.7 g D2O was given per kilogram of body
  water and deuterium incorporation into the erythrocyte cholesterol
  pool was measured for 24 hours. Urinary mevalonate levels were also
  determined in samples obtained during two consecutive 24-hour periods.
  Both techniques provided measurements of whole-body cholesterol
  biosynthesis. In all subjects the cholesterol synthesis rate as
  measured by deuterium incorporation was significantly lower (P&lt;.05)
  after the transition from low- to medium- and low- to high-cholesterol
  diets. Urinary mevalonate excretion decreased after the change from
  the medium- to high- (P&lt;.05) and low- to high- (P&lt;.01) cholesterol
  diets. Although correspondence between the two methods was poor, they
  both indicated some suppression of cholesterol synthesis by dietary
  cholesterol. The response of cholesterogenesis to different amounts of
  dietary cholesterol was related to the rate of synthesis under
  depressed conditions of the low-cholesterol diet. These findings
  indicate modest downregulation of synthesis in response to dietary
  cholesterol in humans, independent of plasma cholesterol levels.
  
  Metabolic responses to increased dietary cholesterol potentially
  include reduced endogenous synthesis, decreased absorption, and
  increased biliary excretion of cholesterol.


In some people, who are cholesterol hyperresponders, dietary cholesterol does not effectively suppress endogenous cholesterol synthesis, so it results in elevated blood cholesterol levels (Comprehensive Biotechnology, 2011). In one study, a decrease in dietary cholesterol intake resulted in a decrease of blood cholesterol in hyperresponders but not in hyporesponders: Endogenous cholesterol synthesis, fecal steroid excretion and serum lanosterol in subjects with high or low response of serum cholesterol to dietary cholesterol (Clinical Nutrition, 1986):


  In this study we addressed the question whether hypo- and
  hyper-responders to dietary cholesterol differ with regard to the
  flexibility of endogenous cholesterol synthesis after changes in
  cholesterol intake. Whole-body cholesterol synthesis was measured as
  faecal excretion of neutral steroids and bile acids minus cholesterol
  intake. In addition, we determined serum concentrations of lanosterol,
  a precursor of cholesterol and a possible indicator of cholesterol
  biosynthetic activity. The study was carried out with 2 hyper- and 4
  hypo-responders; these subjects had shown a consistently high or low
  response of serum cholesterol to a decrease in dietary cholesterol in
  two previous experiments.
  
  The subjects received controlled high- (on average 697 mg of
  cholesterol per day) and low-cholesterol (109 mg/day) diets for
  periods of 4 weeks in succession; cholesterol was the only dietary
  variable. The two hyper-responders again showed a significant decrease
  in serum cholesterol. There was essentially no decrease in serum
  cholesterol in three of the four hypo-responders. The decrease in
  cholesterol intake caused an increase in cholesterol synthesis in five
  out of the six subjects.


In conclusion, in most people, dietary intake of cholesterol does not increase blood cholesterol levels because of decreased absorption and endogenous synthesis and increased excretion of cholesterol.