  We study two classic variants of block-structured integer programming.
Two-stage stochastic programs are integer programs of the form $\{A_i
\mathbf{x} + D_i \mathbf{y}_i = \mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$,
where $A_i$ and $D_i$ are bounded-size matrices. On the other hand, $n$-fold
programs are integer programs of the form $\{{\sum_{i=1}^n
C_i\mathbf{y}_i=\mathbf{a}} \textrm{ and } D_i\mathbf{y}_i=\mathbf{b}_i\textrm{
for all }i=1,\ldots,n\}$, where again $C_i$ and $D_i$ are bounded-size
matrices. It is known that solving these kind of programs is fixed-parameter
tractable when parameterized by the maximum dimension among the relevant
matrices $A_i,C_i,D_i$ and the maximum absolute value of any entry appearing in
the constraint matrix.
  We show that the parameterized tractability results for two-stage stochastic
and $n$-fold programs persist even when one allows large entries in the global
part of the program. More precisely, we prove that:
  - The feasibility problem for two-stage stochastic programs is
fixed-parameter tractable when parameterized by the dimensions of matrices
$A_i,D_i$ and by the maximum absolute value of the entries of matrices $D_i$.
That is, we allow matrices $A_i$ to have arbitrarily large entries.
  - The linear optimization problem for $n$-fold integer programs that are
uniform -- all matrices $C_i$ are equal -- is fixed-parameter tractable when
parameterized by the dimensions of matrices $C_i$ and $D_i$ and by the maximum
absolute value of the entries of matrices $D_i$. That is, we require that
$C_i=C$ for all $i=1,\ldots,n$, but we allow $C$ to have arbitrarily large
entries.
  In the second result, the uniformity assumption is necessary; otherwise the
problem is $\mathsf{NP}$-hard already when the parameters take constant values.
Both our algorithms are weakly polynomial: the running time is measured in the
total bitsize of the input.
