How to flatten / merge overlapping time periods

Here's a possible solution. The basic idea here is to compare lagged start date with the maximum end date "until now" using the cummax function and create an index that will separate the data into groups

data %>%
  arrange(ID, start) %>% # as suggested by @Jonno in case the data is unsorted
  group_by(ID) %>%
  mutate(indx = c(0, cumsum(as.numeric(lead(start)) >
                     cummax(as.numeric(end)))[-n()])) %>%
  group_by(ID, indx) %>%
  summarise(start = first(start), end = last(end))

# Source: local data frame [3 x 4]
# Groups: ID
# 
#   ID indx      start        end
# 1  A    0 2013-01-01 2013-01-06
# 2  A    1 2013-01-07 2013-01-11
# 3  A    2 2013-01-12 2013-01-15

@David Arenburg's answer is great - but I ran into an issue where an earlier interval ended after a later interval - but using last in the summarise call resulted in the wrong end date. I'd suggest changing first(start) and last(end) to min(start) and max(end)

data %>%
  group_by(ID) %>%
  mutate(indx = c(0, cumsum(as.numeric(lead(start)) >
                     cummax(as.numeric(end)))[-n()])) %>%
  group_by(ID, indx) %>%
  summarise(start = min(start), end = max(end))

Also, as @Jonno Bourne mentioned, sorting by start and any grouping variables is important before applying the method.